55 inequalities and comparative statements
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Transcript of 55 inequalities and comparative statements
![Page 1: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/1.jpg)
Inequalities
![Page 2: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/2.jpg)
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
Inequalities
![Page 3: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/3.jpg)
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
Inequalities
![Page 4: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/4.jpg)
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3
Inequalities
![Page 5: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/5.jpg)
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½
Inequalities
![Page 6: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/6.jpg)
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
Inequalities
–π –3.14..
![Page 7: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/7.jpg)
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
This line with each position addressed by a real number is
called the real (number) line.
Inequalities
–π –3.14..
![Page 8: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/8.jpg)
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
This line with each position addressed by a real number is
called the real (number) line.
Inequalities
–π –3.14..
Given two numbers corresponding to two points on the real
line, we define the number to the right to be greater than the
number to the left.
![Page 9: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/9.jpg)
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
This line with each position addressed by a real number is
called the real (number) line.
Inequalities
+–RL
–π –3.14..
Given two numbers corresponding to two points on the real
line, we define the number to the right to be greater than the
number to the left.
![Page 10: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/10.jpg)
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
This line with each position addressed by a real number is
called the real (number) line.
Inequalities
+–R
We write this as L < R and called this the natural form because
it corresponds to their respective positions on the real line.
L <
–π –3.14..
Given two numbers corresponding to two points on the real
line, we define the number to the right to be greater than the
number to the left.
![Page 11: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/11.jpg)
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
This line with each position addressed by a real number is
called the real (number) line.
Inequalities
+–R
We write this as L < R and called this the natural form because
it corresponds to their respective positions on the real line.
This relation may also be written as R > L (less preferable).
L <
–π –3.14..
Given two numbers corresponding to two points on the real
line, we define the number to the right to be greater than the
number to the left.
![Page 12: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/12.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
Inequalities
![Page 13: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/13.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
![Page 14: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/14.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x".
![Page 15: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/15.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a).
![Page 16: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/16.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
+–a
open dot
a < x
![Page 17: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/17.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
+–a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x.
a < x
![Page 18: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/18.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
+–a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
+–a
solid dot
a < x
a < x
![Page 19: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/19.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
+–a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
+–a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all
the numbers x between a and b.
![Page 20: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/20.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
+–a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
+–a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all
the numbers x between a and b.
+–a a < x < b b
![Page 21: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/21.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
+–a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
+–a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all
the numbers x between a and b. A line segment as such is
called an interval.
+–a a < x < b b
![Page 22: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/22.jpg)
Example B.
a. Draw –1 < x < 3.
Inequalities
![Page 23: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/23.jpg)
Example B.
a. Draw –1 < x < 3.
Inequalities
It’s in the natural form.
![Page 24: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/24.jpg)
Example B.
a. Draw –1 < x < 3.
Inequalities
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
![Page 25: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/25.jpg)
Example B.
a. Draw –1 < x < 3.
0 3+
-1– x
Inequalities
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
![Page 26: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/26.jpg)
Example B.
a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
Inequalities
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
![Page 27: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/27.jpg)
Example B.
a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
Inequalities
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
Put it in the natural form –3 < x < 0.
![Page 28: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/28.jpg)
Example B.
a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
Inequalities
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
![Page 29: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/29.jpg)
Example B.
a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Inequalities
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
![Page 30: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/30.jpg)
Example B.
a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Expressions such as 2 < x > 3 or 2 < x < –3 do not have
any solution.
Inequalities
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
![Page 31: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/31.jpg)
Example B.
a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Expressions such as 2 < x > 3 or 2 < x < –3 do not have
any solution.
Inequalities
Adding or subtracting the same quantity to both retains the
inequality sign,
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
![Page 32: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/32.jpg)
Example B.
a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Expressions such as 2 < x > 3 or 2 < x < –3 do not have
any solution.
Inequalities
Adding or subtracting the same quantity to both retains the
inequality sign, i.e. if a < b, then a ± c < b ± c.
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
![Page 33: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/33.jpg)
Example B.
a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Expressions such as 2 < x > 3 or 2 < x < –3 do not have
any solution.
Inequalities
For example 6 < 12, then 6 + 3 < 12 + 3.
Adding or subtracting the same quantity to both retains the
inequality sign, i.e. if a < b, then a ± c < b ± c.
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
![Page 34: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/34.jpg)
Example B.
a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Expressions such as 2 < x > 3 or 2 < x < –3 do not have
any solution.
Inequalities
For example 6 < 12, then 6 + 3 < 12 + 3.
We use the this fact to solve inequalities.
Adding or subtracting the same quantity to both retains the
inequality sign, i.e. if a < b, then a ± c < b ± c.
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
![Page 35: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/35.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
Inequalities
![Page 36: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/36.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides
x – 3 + 3 < 12 + 3
Inequalities
![Page 37: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/37.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
Inequalities
![Page 38: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/38.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
0 15+–
Inequalities
x
![Page 39: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/39.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c.
![Page 40: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/40.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign,
![Page 41: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/41.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b
![Page 42: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/42.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc.
![Page 43: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/43.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc. For example 6 < 12 is true,
![Page 44: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/44.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc. For example 6 < 12 is true, then multiplying by 3
3*6 < 3*12
![Page 45: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/45.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc. For example 6 < 12 is true, then multiplying by 3
3*6 < 3*12 or 18 < 36 is also true.
![Page 46: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/46.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc.
Example D. Solve 3x > 12 and draw the solution.
For example 6 < 12 is true, then multiplying by 3
3*6 < 3*12 or 18 < 36 is also true.
![Page 47: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/47.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc.
Example D. Solve 3x > 12 and draw the solution.
3x > 12
For example 6 < 12 is true, then multiplying by 3
3*6 < 3*12 or 18 < 36 is also true.
![Page 48: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/48.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc.
Example D. Solve 3x > 12 and draw the solution.
3x > 12 divide by 3 and keep the inequality sign 3x/3 > 12/3
For example 6 < 12 is true, then multiplying by 3
3*6 < 3*12 or 18 < 36 is also true.
![Page 49: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/49.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc.
Example D. Solve 3x > 12 and draw the solution.
3x > 12 divide by 3 and keep the inequality sign 3x/3 > 12/3
x > 4 or 4 < x
For example 6 < 12 is true, then multiplying by 3
3*6 < 3*12 or 18 < 36 is also true.
![Page 50: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/50.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc.
Example D. Solve 3x > 12 and draw the solution.
3x > 12 divide by 3 and keep the inequality sign 3x/3 > 12/3
x > 4 or 4 < x
40+–
For example 6 < 12 is true, then multiplying by 3
3*6 < 3*12 or 18 < 36 is also true.
x
![Page 51: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/51.jpg)
A number c is negative means c < 0.
Inequalities
![Page 52: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/52.jpg)
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
Inequalities
![Page 53: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/53.jpg)
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
Inequalities
![Page 54: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/54.jpg)
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
Inequalities
![Page 55: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/55.jpg)
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
Inequalities
For example 6 < 12 is true.
![Page 56: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/56.jpg)
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
Inequalities
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
![Page 57: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/57.jpg)
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
Inequalities
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
![Page 58: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/58.jpg)
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
Inequalities
60+–
12<
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
![Page 59: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/59.jpg)
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
Inequalities
60+–
12–6 <
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
![Page 60: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/60.jpg)
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
![Page 61: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/61.jpg)
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
![Page 62: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/62.jpg)
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
–x + 2 < 5
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
![Page 63: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/63.jpg)
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
–x + 2 < 5 subtract 2 from both sides
–x < 3
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
![Page 64: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/64.jpg)
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
–x + 2 < 5 subtract 2 from both sides
–x < 3 multiply by –1 to get x, reverse the inequality
–(–x) > –3
x > –3
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
![Page 65: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/65.jpg)
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
–x + 2 < 5 subtract 2 from both sides
–x < 3 multiply by –1 to get x, reverse the inequality
–(–x) > –3
x > –3 or –3 < x
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
![Page 66: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/66.jpg)
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
–x + 2 < 5 subtract 2 from both sides
–x < 3 multiply by –1 to get x, reverse the inequality
–(–x) > –3
x > –3 or –3 < x
0+
-3–
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
![Page 67: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/67.jpg)
To solve inequalities:
1. Simplify both sides of the inequalities
Inequalities
![Page 68: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/68.jpg)
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides
Inequalities
![Page 69: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/69.jpg)
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
Inequalities
![Page 70: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/70.jpg)
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x.
Inequalities
![Page 71: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/71.jpg)
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around.
Inequalities
![Page 72: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/72.jpg)
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Inequalities
![Page 73: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/73.jpg)
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
![Page 74: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/74.jpg)
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign
![Page 75: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/75.jpg)
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign
3x – x > 9 – 5
![Page 76: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/76.jpg)
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign
3x – x > 9 – 5
2x > 4
![Page 77: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/77.jpg)
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign
3x – x > 9 – 5
2x > 4 div. 2
2x 2
4 2>
![Page 78: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/78.jpg)
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign
3x – x > 9 – 5
2x > 4 div. 2
2x 2
4 2>
x > 2 or 2 < x
![Page 79: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/79.jpg)
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign
3x – x > 9 – 5
2x > 4 div. 2
20+–
2x 2
4 2>
x > 2 or 2 < x
![Page 80: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/80.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
Inequalities
![Page 81: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/81.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side
Inequalities
![Page 82: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/82.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x
Inequalities
![Page 83: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/83.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18
Inequalities
![Page 84: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/84.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x
Inequalities
![Page 85: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/85.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x
– 12 > 3x
Inequalities
![Page 86: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/86.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x
– 12 > 3x
–12 3
3x 3
>
div. by 3 (no need to switch >)
Inequalities
![Page 87: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/87.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x
– 12 > 3x
–12 3
3x 3
>
–4 > x or x < –4
div. by 3 (no need to switch >)
Inequalities
![Page 88: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/88.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x
– 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
–4 > x or x < –4
![Page 89: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/89.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x
– 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
We also have inequalities in the form of intervals.
–4 > x or x < –4
![Page 90: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/90.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x
– 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.
–4 > x or x < –4
![Page 91: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/91.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x
– 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.
Again, we + or – remove the number term in the middle first,
–4 > x or x < –4
![Page 92: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/92.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x
– 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.
Again, we + or – remove the number term in the middle first,
then divide or multiply to get x.
–4 > x or x < –4
![Page 93: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/93.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x
– 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.
Again, we + or – remove the number term in the middle first,
then divide or multiply to get x. The answer is an interval of
numbers.
–4 > x or x < –4
![Page 94: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/94.jpg)
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw
Inequalities
![Page 95: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/95.jpg)
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6 –6 –6 –6
Inequalities
![Page 96: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/96.jpg)
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10
Inequalities
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Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10 div. by -2, switch inequality sign 6 -2
-2x
-2<
-10
-2<
Inequalities
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Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10
-3 < x < 5
div. by -2, switch inequality sign 6 -2
-2x
-2<
-10
-2<
Inequalities
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Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10
0+
-3 < x < 5
5
div. by -2, switch inequality sign 6 -2
-2x
-2<
-10
-2<
-3
Inequalities
The following adjectives or comparison phrases may be
translated into inequalities:
![Page 100: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/100.jpg)
The following adjectives or comparison phrases may be
translated into inequalities:
“positive”↔“negative”,
“non–positive”↔”non–negative”,
“more/greater than”↔ “less/smaller than”,
“no more/greater than”↔ “no less/smaller than”,
“ at least” ↔”at most”,
Inequalities and Comparative Phrases
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The following adjectives or comparison phrases may be
translated into inequalities:
“positive”↔“negative”,
“non–positive”↔”non–negative”,
“more/greater than”↔ “less/smaller than”,
“no more/greater than”↔ “no less/smaller than”,
“ at least” ↔”at most”,
Positive vs. Negative”
Inequalities and Comparative Phrases
![Page 102: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/102.jpg)
The following adjectives or comparison phrases may be
translated into inequalities:
“positive”↔“negative”,
“non–positive”↔”non–negative”,
“more/greater than”↔ “less/smaller than”,
“no more/greater than”↔ “no less/smaller than”,
“ at least” ↔”at most”,
A quantity x is positive means that x is more than 0, i.e. 0 < x,
Positive vs. Negative”
Inequalities and Comparative Phrases
![Page 103: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/103.jpg)
The following adjectives or comparison phrases may be
translated into inequalities:
“positive”↔“negative”,
“non–positive”↔”non–negative”,
“more/greater than”↔ “less/smaller than”,
“no more/greater than”↔ “no less/smaller than”,
“ at least” ↔”at most”,
A quantity x is positive means that x is more than 0, i.e. 0 < x,
0
+x is positive
Positive vs. Negative”
On the real line:
Inequalities and Comparative Phrases
![Page 104: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/104.jpg)
The following adjectives or comparison phrases may be
translated into inequalities:
“positive”↔“negative”,
“non–positive”↔”non–negative”,
“more/greater than”↔ “less/smaller than”,
“no more/greater than”↔ “no less/smaller than”,
“ at least” ↔”at most”,
A quantity x is positive means that x is more than 0, i.e. 0 < x,
and that x is negative means that x is less than 0, i.e. x < 0.
0
+– x is negative x is positive
Positive vs. Negative”
On the real line:
Inequalities and Comparative Phrases
![Page 105: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/105.jpg)
The following adjectives or comparison phrases may be
translated into inequalities:
“positive”↔“negative”,
“non–positive”↔”non–negative”,
“more/greater than”↔ “less/smaller than”,
“no more/greater than”↔ “no less/smaller than”,
“ at least” ↔”at most”,
A quantity x is positive means that x is more than 0, i.e. 0 < x,
and that x is negative means that x is less than 0, i.e. x < 0.
0
+– x is negative x is positive
Hence “the temperature T is positive” is translated as “0 < T”.
“the account balance A is negative”, is translated as “A < 0”.
Positive vs. Negative”
On the real line:
Inequalities and Comparative Phrases
![Page 106: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/106.jpg)
The following adjectives or comparison phrases may be
translated into inequalities:
“positive”↔“negative”,
“non–positive”↔”non–negative”,
“more/greater than”↔ “less/smaller than”,
“no more/greater than”↔ “no less/smaller than”,
“ at least” ↔”at most”,
A quantity x is positive means that x is more than 0, i.e. 0 < x,
and that x is negative means that x is less than 0, i.e. x < 0.
0
+– x is negative x is positive
Hence “the temperature T is positive” is translated as “0 < T”.
“the account balance A is negative”, is translated as “A < 0”.
Positive vs. Negative”
On the real line:
Inequalities and Comparative Phrases
![Page 107: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/107.jpg)
Inequalities and Comparative PhrasesNon–Positive vs. Non–Negative
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A quantity x is non–positive means x is not positive, or “x ≤ 0”,
and that x is non–negative means x is not negative, or “0 ≤ x”,
Non–Positive vs. Non–Negative
Inequalities and Comparative Phrases
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A quantity x is non–positive means x is not positive, or “x ≤ 0”,
and that x is non–negative means x is not negative, or “0 ≤ x”,
0
+
– x is non–positive
x is non–negative
Non–Positive vs. Non–Negative
On the real line:
Inequalities and Comparative Phrases
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A quantity x is non–positive means x is not positive, or “x ≤ 0”,
and that x is non–negative means x is not negative, or “0 ≤ x”,
0
+
– x is non–positive
x is non–negative
Hence “the temperature T is non–positive” is “T ≤ 0”.
“the account balance A is non–negative” is “0 ≤ A”.
Non–Positive vs. Non–Negative
On the real line:
Inequalities and Comparative Phrases
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A quantity x is non–positive means x is not positive, or “x ≤ 0”,
and that x is non–negative means x is not negative, or “0 ≤ x”,
0
+
– x is non–positive
x is non–negative
Hence “the temperature T is non–positive” is “T ≤ 0”.
“the account balance A is non–negative” is “0 ≤ A”.
Non–Positive vs. Non–Negative
On the real line:
“More/greater than” vs “Less/smaller than”
Inequalities and Comparative Phrases
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A quantity x is non–positive means x is not positive, or “x ≤ 0”,
and that x is non–negative means x is not negative, or “0 ≤ x”,
0
+
– x is non–positive
x is non–negative
Hence “the temperature T is non–positive” is “T ≤ 0”.
“the account balance A is non–negative” is “0 ≤ A”.
Non–Positive vs. Non–Negative
On the real line:
“More/greater than” vs “Less/smaller than”
Let C be a number, x is greater than C means “C < x”,
and that x is less than C means “x < C”.
Inequalities and Comparative Phrases
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A quantity x is non–positive means x is not positive, or “x ≤ 0”,
and that x is non–negative means x is not negative, or “0 ≤ x”,
0
+
– x is non–positive
x is non–negative
Hence “the temperature T is non–positive” is “T ≤ 0”.
“the account balance A is non–negative” is “0 ≤ A”.
Non–Positive vs. Non–Negative
On the real line:
“More/greater than” vs “Less/smaller than”
Let C be a number, x is greater than C means “C < x”,
and that x is less than C means “x < C”.
C
x is less than C
x is more than COn the real line:
Inequalities and Comparative Phrases
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A quantity x is non–positive means x is not positive, or “x ≤ 0”,
and that x is non–negative means x is not negative, or “0 ≤ x”,
0
+
– x is non–positive
x is non–negative
Hence “the temperature T is non–positive” is “T ≤ 0”.
“the account balance A is non–negative” is “0 ≤ A”.
Non–Positive vs. Non–Negative
On the real line:
“More/greater than” vs “Less/smaller than”
Let C be a number, x is greater than C means “C < x”,
and that x is less than C means “x < C”.
C
x is less than C
x is more than C
Hence “the temperature T is more than –5 ” is “–5 < T”.
“the account balance A is less than 1,000” is “A < 1,000”.
On the real line:
Inequalities and Comparative Phrases
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“ No more/greater than” vs “No less/smaller than”
Inequalities and Comparative Phrases
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“ No more/greater than” vs “No less/smaller than”
A quantity x is no more greater than C means “x ≤ C”,
and that x is no–less than C means “C ≤ x”,
Inequalities and Comparative Phrases
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“ No more/greater than” vs “No less/smaller than”
A quantity x is no more greater than C means “x ≤ C”,
and that x is no–less than C means “C ≤ x”,
0
+
– x is no more than C
x is no less than COn the real line:C
Inequalities and Comparative Phrases
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“ No more/greater than” vs “No less/smaller than”
A quantity x is no more greater than C means “x ≤ C”,
and that x is no–less than C means “C ≤ x”,
0
+
– x is no more than C
x is no less than C
“The temperature T is no–more than 250o” is “T ≤ 250”.
“The account balance A is no–less than 500” is “500 ≤ A”.
On the real line:C
Inequalities and Comparative Phrases
![Page 119: 55 inequalities and comparative statements](https://reader033.fdocuments.in/reader033/viewer/2022051404/58ed57a01a28ab32178b4569/html5/thumbnails/119.jpg)
“ No more/greater than” vs “No less/smaller than”
A quantity x is no more greater than C means “x ≤ C”,
and that x is no–less than C means “C ≤ x”,
0
+
– x is no more than C
x is no less than C
“The temperature T is no–more than 250o” is “T ≤ 250”.
“The account balance A is no–less than 500” is “500 ≤ A”.
On the real line:C
“At least” vs “At most”
“At least C” is the same as “no less than C” and
“at most C” means “no more than C”.
Inequalities and Comparative Phrases
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“ No more/greater than” vs “No less/smaller than”
A quantity x is no more greater than C means “x ≤ C”,
and that x is no–less than C means “C ≤ x”,
0
+
– x is no more than C
x is no less than C
“The temperature T is no–more than 250o” is “T ≤ 250”.
“The account balance A is no–less than 500” is “500 ≤ A”.
On the real line:C
“At least” vs “At most”
“At least C” is the same as “no less than C” and
“at most C” means “no more than C”.
0
+
– x is at most C
x is at least COn the real line:
C
Inequalities and Comparative Phrases
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“ No more/greater than” vs “No less/smaller than”
A quantity x is no more greater than C means “x ≤ C”,
and that x is no–less than C means “C ≤ x”,
0
+
– x is no more than C
x is no less than C
“The temperature T is no–more than 250o” is “T ≤ 250”.
“The account balance A is no–less than 500” is “500 ≤ A”.
On the real line:C
“At least” vs “At most”
“At least C” is the same as “no less than C” and
“at most C” means “no more than C”.
0
+
– x is at most C
x is at least C
“The temperature T is at least 250o” is “250o ≤ T”.
“The account balance A is at most than 500” is “A ≤ 500”.
On the real line:
C
Inequalities and Comparative Phrases
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InequalitiesExercise. A. Draw the following Inequalities. Indicate clearly
whether the end points are included or not.
1. x < 3 2. –5 ≤ x 3. x < –8 4. x ≤ 12
B. Write in the natural form then draw them.
5. x ≥ 3 6. –5 > x 7. x ≥ –8 8. x > 12
C. Draw the following intervals, state so if it is impossible.
9. 6 > x ≥ 3 10. –5 < x ≤ 2 11. 1 > x ≥ –8 12. 4 < x ≤ 2
13. 6 > x ≥ 8 14. –5 > x ≤ 2 15. –7 ≤ x ≤ –3 16. –7 ≤ x ≤ –9
D. Solve the following Inequalities and draw the solution.
17. x + 5 < 3 18. –5 ≤ 2x + 3 19. 3x + 1 < –8
20. 2x + 3 ≤ 12 – x 21. –3x + 5 ≥ 1 – 4x
22. 2(x + 2) ≥ 5 – (x – 1) 23. 3(x – 1) + 2 ≤ – 2x – 9
24. –2(x – 3) > 2(–x – 1) + 3x 25. –(x + 4) – 2 ≤ 4(x – 1)
26. x + 2(x – 3) < 2(x – 1) – 2
27. –2(x – 3) + 3 ≥ 2(x – 1) + 3x + 13
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Inequalities
F. Solve the following interval inequalities.
28. –4 ≤2x
29. 7 >3–x
30. < –4–x
E. Clear the denominator first then solve and draw the solution.
5
x 2 31 2
32
+ ≥ x31. x 4
–3 3
–4– 1 > x32.
x 2 83 3
45
– ≤33. x 8 12
–5 71 + > 34.
x 2 3
–3 234
41
–+ x35. x 4 65 5
3–1
– 2 + < x36.
x 12 27 3
61
43
–– ≥ x37.
≤
40. – 2 < x + 2 < 5 41. –1 ≥ 2x – 3 ≥ –11
42. –5 ≤ 1 – 3x < 10 43. 11 > –1 – 3x > –7
38. –6 ≤ 3x < 12 39. 8 > –2x > –4
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Exercise. G. Draw the following Inequalities. Translate each
inequality into an English phrase. (There might be more than
one ways to do it)
1. x < 3 2. –5 ≤ x 3. x < –8 4. x ≤ 12
5. x ≥ 3 6. –5 > x 7. x ≥ –8 8. x > 12
Exercise. B. Translate each English phrase into an inequality.
Draw the Inequalities.
Let P be the number of people on a bus.
9. There were at least 50 people on the bus.
10. There were no more than 50 people on the bus.
11. There were less than 30 people on the bus.
12. There were no less than 28 people on the bus.
Let T be temperature outside.
13. The temperature is no more than –2o.
14. The temperature is at least than 35o.
15. The temperature is positive.
Inequalities and Comparative Phrases
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Let M be the amount of money I have.
16. I have at most $25.
17. I have non–positive amount of money.
18. I have less than $45.
19. I have at least $250.
Let the basement floor number be given as negative number
and that F be floor number that we are on.
20. We are below the 7th floor.
21. We are above the first floor.
22. We are not below the 3rd floor basement.
24. We are on at least the 45th floor.
25. We are between the 4th floor basement and the 10th
floor.26. We are in the basement.
Inequalities and Comparative Phrases