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![Page 1: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/1.jpg)
Solving Inequalities
![Page 2: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/2.jpg)
Solving Inequalities
3
2
Example 1
932 x
These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow most of the same rules.3
62 x2
3x2
3 4
Work the problem out as if it has an equals sign. Then when you get an answer, sketch the graph on the number line and write your answer in interval notation.
![Page 3: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/3.jpg)
Solving Inequalities
x3
4
Example 2
8325 xxWhy is this different from Example 1? Because it has an x on both sides of the inequality. However, it doesn’t change how you work the problem.
x3822 x
2102 x2
5 6
2 25x
![Page 4: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/4.jpg)
Solving Inequalities
9
5
Example 3
2139 xThis is the first example of a problem that is a little different than solving equations.
9123 x
4 3
3 34x
Anytime you multiply or divide by a negative, you have to turn the inequality symbol around.
![Page 5: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/5.jpg)
Solving Inequalities
x3 x3
Example 4
2343 xx For this problem, since the variables subtracted off, it’s either “no solution” or “all real numbers”.It is “no solution” if the statement leftover is false.It is “all real numbers” if the statement is true.
24
![Page 6: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/6.jpg)
Solving Inequalities
x2
1
Example 5
)4(2)2(4 xxIs this “no solution” or “all real numbers”?Neither. It’s only one of those if the variables go away.
x2882 x
802 x8
0 1
2 20x
8284 xx
![Page 7: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/7.jpg)
Solving Rational Equations
x5
3067 x
247 x
Example 6
53
5
3
2
xx Remember, to multiply fractions
times whole numbers, divide the bottom into the whole number, then multiply what’s left.
( )
10
6 30 x2x2x2
6 6
7
24x
7
24
7
237
25
![Page 8: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/8.jpg)
Solving Inequalities
]2
25,()4
)3
),7[)2
)3,)(1
Practice:
354
3
2)4
)23(4)14(3)3
15865)2
1024)1
xx
xx
xx
x
Answers:
![Page 9: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/9.jpg)
Solving Absolute Value Equations
![Page 10: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/10.jpg)
Solving Absolute Value Equations
53 x
Example 1
53 xThis problem has absolute value bars in it. Anytime you see absolute value bars in an equation, you need to split the problem into two different problems. The first equation is the exact as the original except just erase the absolute value bars. For the second equation, just change the sign of the other side.
53 x3 3 3
2x38x
OR
![Page 11: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/11.jpg)
Solving Absolute Value Equations
1563 x
Example 2
1563 x1563 x
6 6 6
3
93
x
x
6
7
213
x
x
OR
![Page 12: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/12.jpg)
Solving Absolute Value Equations
1312 x1312 x
4
Example 3
9412 x Always make sure that the absolute value bars are alone first, so add the four to both sides before you split it into two.
1312 x1 1
7x
1
6x
OR
4
1142 x
2 2122 x
2 2
![Page 13: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/13.jpg)
Solving Absolute Value Equations
32 x32 x
3
Example 3.5
923 x You cannot distribute numbers into the absolute values. Since the negative three is being multiplied times the absolute value bars, to get rid of them, we need to divide both sides by the negative three.
32 x2 2
5x
2
1x
OR
3
2
![Page 14: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/14.jpg)
Solving Absolute Value Equations
Example 4
34 xBecause the absolute values can never equal a negative, there is no work involved on this problem.
![Page 15: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/15.jpg)
Solving Absolute Value Equations
x2 x2xx 92
Example 5
xx 92 Whenever there are variables inside the absolute value bars AND outside, you HAVE TO CHECK YOUR ANSWERS!!!
x2
x9
x2
x3
ORxx 92
x 91 1
x39 3 3
![Page 16: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/16.jpg)
Solving Absolute Value Equations
918
9
39)3(2
Example 5
xx 92 Remember the rules for checking. 1) Always go back to the original problem.2) Do not cross the equals sign.
3) Use order of operations on each side.
96
3
Hurray! Both of them worked.
99)9(2
9 3
xx 92
![Page 17: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/17.jpg)
Solving Absolute Value Equations
x xxx 212
Example 6
xx 212 Do I have to check my answers here?
x x
x4
ORxx 212
x 12 x312 3 3
![Page 18: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/18.jpg)
Solving Absolute Value Equations
2424
2424
)4(2124
Example 6
xx 212
88
88 Since this gives you a true statement, 4 works. Therefore, x = 4.
)12(21212
xx 212
![Page 19: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/19.jpg)
Solving Absolute Value Equations
x xxx 212
Example 6
xx 212 How do you communicate that?Mark off the answers that don’t work, and circle the ones that do.What is your answer if NEITHER answer checks?
x x
x4
ORxx 212
x 12 x312 3 3
![Page 20: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/20.jpg)
Solving Absolute Value Equations
x x392 xx
Example 7
392 xx
x x
2x
OR392 xx
39 x 393 x
33
9 9
12x9 9
63 x
![Page 21: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/21.jpg)
Solving Absolute Value Equations
15924
1515
329)2(2
Example 7
392 xx
594
55
Since neither one works, there are NO SOLUTIONS.
3129)12(2
1515 55
392 xx
![Page 22: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/22.jpg)
Solving Absolute Value Inequalities and Compound Inequalities
![Page 23: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/23.jpg)
Solving Absolute Value Inequalities
or
7),3()7,(
Example 6
52 x Because there is an absolute value in the problem, that tells me that I have to split the problem into two pieces.When you write it the second time, not only do you change the sign, but you also turn the inequality around.To decide if you use “and” or “or”, remember GO to LA. Greater thanOrLess thanAnd
52 x2
3x 7x
0 3
52 x2 2 2
With “or”, just put both inequalities on the final
graph.
![Page 24: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/24.jpg)
Solving Absolute Value Inequalities
and
4)8,4(
Example 7
62 x Because there is an absolute value in the problem, that tells me that I have to split the problem into two pieces.To decide if you use “and” or “or”, remember GO to LA. Greater thanOrLess thanAndWhen you write it the second time, not only do you change the sign, but you also turn the inequality around.
62 x2
8x 4x
0 8
62 x2 2 2
With “and”, find where the two inequalities
intersect, and put that on the final graph.
![Page 25: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/25.jpg)
Solving Absolute Value Inequalities
or
2),6[]2,(
Example 8
842 x842 x
122 x
0 6
842 x
6x 2x42 x
![Page 26: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/26.jpg)
Solving Absolute Value Inequalities
Example 9
845 x When there is a negative on the other side of an absolute value inequality, the answer is either “no solution” or “all real numbers”.Because the absolute value will always be positive, if it is a greater than, it will be “all real numbers”. If there is a less than sign with the negative on the outside, the answer is “no solution”.
273 xExample 10
![Page 27: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/27.jpg)
Solving Absolute Value Inequalities
and
2]6,2[
Example 11
1236 x1236 x
63 x
0 6
1236 x
2x 6x183 x
![Page 28: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/28.jpg)
Solving Inequalities
3
2
),3(
Example 12
932 x
YOU DO NOT BREAK THIS INTO TWO PROBLEMS BECAUSE THERE ARE NO ABSOLUTE VALUE BARS!!!
362 x
23x2
3 4
![Page 29: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/29.jpg)
Solving Absolute Value Inequalities
3),8[)3,(
Example 13
17or 63 xxThis is a compound inequality. It is already set up to start solving the separate equations.Since it has an “or” between the two, just put both graphs on the final graph and write your answer in interval notation.
33x 8x
5 8
3 7 7
![Page 30: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/30.jpg)
Solving Absolute Value Inequalities
3)4,3[
1239 x
Example 14
93312 x This is another type of compound inequality. Whatever you do to get the x by itself in the middle, you have to do it to all “sides” of the inequality.Since it is written with two inequalities in one sentence, it is understood to have an “and” between them. Therefore, solve, and find the intersection.
3 3
43 x
0 4
3
3 3 3
![Page 31: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/31.jpg)
Solving Absolute Value Inequalities
4628)4
26)3
15123)2
952)1
x
x
x
x
)1,7)[4
)3
]9,1)[2
),2()7,)(1
Practice. Answers:
![Page 32: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/32.jpg)
Solving Absolute Value Inequalities
),2()2,()20
),25()45,()19
4)18
)17
)16
]5
6,2)[15
),7[]3,)(14
]4,3)[13
]2,6)[12
),1()9,)(11
x
Your turn. Do the worksheet on Inequalities. You may work together. This is for homework if you do not finish during class. Show your work!!!
)10
)9
)2,()8
]5,)(7
),2)[6
]7,)(5
]16,)(4
]3,)(3
)5,)(2
),2)(1
![Page 33: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/33.jpg)
For Thursday and Monday…
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2nd Block
![Page 34: Solving Inequalities. Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow.](https://reader036.fdocuments.in/reader036/viewer/2022062518/5697bf7b1a28abf838c8366d/html5/thumbnails/34.jpg)
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4th Block