You already know inequalities. Often they are used to place limits on variables. That just means x...
-
Upload
norman-phelps -
Category
Documents
-
view
215 -
download
0
Transcript of You already know inequalities. Often they are used to place limits on variables. That just means x...
![Page 1: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/1.jpg)
x 12
9x
x 12
9x
x 12
9x
x 129x
912 x
![Page 2: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/2.jpg)
9x
You already know inequalities.
Often they are used to place limits on variables.
That just means x can be any number equal to 9 or less than
9.
![Page 3: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/3.jpg)
Sometimes we put more than one limit on the variable:
Now x is still less than or equal to 9, but it must also be greater than or
equal to –7.
7 and 9 xx
![Page 4: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/4.jpg)
7 and 9 xx
Let’s look at the graph:
0 5 10 15-20 -15 -10 -5-25 20 25
The upper limit is 9. Because x can be equal to 9, we mark it with a filled-in
circle.
![Page 5: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/5.jpg)
0 5 10 15-20 -15 -10 -5-25 20 25
The lower limit is -7. We also need to mark it with a filled-in
circle.
7 and 9 xx
![Page 6: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/6.jpg)
Where are they found on the graph?
Yes!
It is less than or equal to 9?What about –15?
0 5 10 15-20 -15 -10 -5-25 20 25
There are other numbers that satisfy both conditions.
7 and 9 xx
15
![Page 7: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/7.jpg)
It is also greater than or equal to -7?
What about –15?
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
15 No!
Where are they found on the graph?
![Page 8: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/8.jpg)
Because the word and is used, a number on the graph
needs to satisfy both parts of the inequality.
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
15
![Page 9: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/9.jpg)
Yes!720
So let’s try 20. Does 20 satisfy both conditions?
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
20
?
15
![Page 10: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/10.jpg)
No!920
So let’s try 20. Does 20 satisfy both conditions?
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
20
?
15
![Page 11: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/11.jpg)
Since 20 does not satisfy both conditions, it can’t
belong to the solution set.
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
2015
![Page 12: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/12.jpg)
There is one region we have not checked.
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
2015
![Page 13: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/13.jpg)
We need to choose a number from that region.
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
2015
You want to choose 0?Good choice! 0 is usually the easiest number to work
with.0
![Page 14: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/14.jpg)
Does 0 satisfy both conditions?
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
2015 0
70 Yes!?
![Page 15: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/15.jpg)
Does 0 satisfy both conditions?
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
2015 0
90 Yes!?
![Page 16: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/16.jpg)
If one number in a region completely satisfies an
inequality,
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
2015 0
you can know that every number in that region
satisfies the inequality.
![Page 17: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/17.jpg)
Let’s graph another inequality:
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
![Page 18: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/18.jpg)
tells us we want an open circle,The first sign
First we mark the boundary points:
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
![Page 19: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/19.jpg)
and the 12 tells us where the circle goes.
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
![Page 20: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/20.jpg)
and the 12 tells us where the circle goes.
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
![Page 21: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/21.jpg)
tells us we want a closed circle,The second sign
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
![Page 22: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/22.jpg)
and the -1 tells us where the circle goes.
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
![Page 23: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/23.jpg)
The boundary points divide the line into three regions:
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
1 2 3
![Page 24: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/24.jpg)
We need to test one point from each region.
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
10 1210
No!? 1210
Yes!?
![Page 25: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/25.jpg)
Notice that the word used is or,
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
10 1210
No! 1210
Yes!
instead of and.
![Page 26: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/26.jpg)
Or means that a number
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
10 1210
No! 1210
Yes!
only needs to meet one condition.
![Page 27: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/27.jpg)
Because –10 meets one condition,
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
1210 1210 Yes!
10
the region to which it belongs . . .. . . belongs to the graph.
![Page 28: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/28.jpg)
Let’s check the next region:
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
125 15 5
No!? No!?
![Page 29: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/29.jpg)
Because –1 meets neither condition,
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
125 15 5
No! No!
the numbers in that regionwill not satisfy the inequality.
![Page 30: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/30.jpg)
Now the final region:
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
1215 115 15
Yes!? No!?
![Page 31: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/31.jpg)
Again, 15 meets one condition
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
1215 115 15
Yes!
so we need to shade that region.
![Page 32: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/32.jpg)
A quick review:
0 5 10 15-20 -15 -10 -5-25 20 25
1. Find and mark the boundary points.2. Test points from each region.3. Shade the regions that satisfy the inequality.? ? ?
To graph a compound inequality:
![Page 33: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/33.jpg)
A quick review:
0 5 10 15-20 -15 -10 -5-25 20 25
1. Find and mark the boundary points.2. Test points from each region.3. Shade the regions that satisfy the inequality. or
![Page 34: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/34.jpg)
Given the graph below, write the inequality.
First, write the boundary points.
0 5 10 15-20 -15 -10 -5-25 20 25
15 20
![Page 35: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/35.jpg)
15
Then look at the marks on the graph,
0 5 10 15-20 -15 -10 -5-25 20 25
20 15 20
and write the correct symbol.
![Page 36: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/36.jpg)
Since x is between the boundary points on the
graph,
15 20
0 5 10 15-20 -15 -10 -5-25 20 25
it will be between the boundary points in the inequality.
xx
![Page 37: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/37.jpg)
20 15 x
Since x is between the boundary points on the
graph,
x0 5 10 15-20 -15 -10 -5-25 20 25
it will be between the boundary points in the inequality.
![Page 38: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/38.jpg)
0 5 10 15-20 -15 -10 -5-25 20 25
Try this one:Again, begin by writing the
boundary points:
10 5
![Page 39: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/39.jpg)
0 5 10 15-20 -15 -10 -5-25 20 25
And again, you need to choose the correct symbols:
10 5
![Page 40: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/40.jpg)
0 5 10 15-20 -15 -10 -5-25 20 25
Because the x-values are not between the boundary points
on the graph,we won’t write x between the
boundary points in the equation.
x x 5 10
![Page 41: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/41.jpg)
0 5 10 15-20 -15 -10 -5-25 20 25
x xx105x
Because the x-values are not between the boundary points
on the graph,we won’t write them between
the boundary points in the equation.
![Page 42: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/42.jpg)
0 5 10 15-20 -15 -10 -5-25 20 25
x xx105x
We will use the word, or, instead:Remember that or means
a number has to satisfy only one of the conditions.
![Page 43: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/43.jpg)
0 5 10 15-20 -15 -10 -5-25 20 25
x x
We will use the word, or, instead:Remember that or means
a number has to satisfy only one of the conditions.
xx 10or 5
![Page 44: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/44.jpg)
0 5 10 15-20 -15 -10 -5-25 20 25
Is there any one number that belongs to both shaded
sections in the graph?
xx 10or 5
Say NO!NO!
![Page 45: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/45.jpg)
0 5 10 15-20 -15 -10 -5-25 20 25
xx 10or 5
So it would be incorrect to use and. And implies that a
number meets both conditions.
![Page 46: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/46.jpg)
. . . you remember that a compound inequality is just two
inequalities put together.
9325 x
Solving compound inequalities is easy if . . .
932 x325 x 9325 x
![Page 47: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/47.jpg)
9325 x
You can solve them both at the same time:
3 3628 x
2 2
3
234 x
![Page 48: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/48.jpg)
510 x
Write the inequality from the graph:
0 5 10 15-20 -15 -10 -5-25 20 25
1: Write boundaries:2: Write signs:3: Write variable:
![Page 49: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/49.jpg)
Solve the inequality:
57415 x
Is this what you did?
771248 x
4
7
4 42 x 3
![Page 50: You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.](https://reader036.fdocuments.in/reader036/viewer/2022062421/56649dbc5503460f94aadef1/html5/thumbnails/50.jpg)
You did remember to reverse the signs . . .
57415 x771248 x
4
7
4 42 x 3
. . . didn’t you?Good job!