1. By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org
Inequalities
By: Jeffrey Bivin
Lake Zurich High School
Last Updated: August 25, 2007

2. Solve for x 3x + 5 < 17 - 5 - 5 3x < 12 3 3 x < 4 4
Interval Notation
3x < 12
x < 4 4
Jeff Bivin -- LZHS

3. Solve for x 3x + 5 < 17 - 5 - 5 3x < 12 3 3 x < 4 4
Interval Notation
3x < 12
x < 4 4
Jeff Bivin -- LZHS

4. Solve for x 11 < 3x + 5 < 26 - 5 - 5 - 5 6 < 3x < 21 3 3 3
Interval Notation
6 < 3x < 21
2 < x < 7 2 7
Jeff Bivin -- LZHS

5. Solve for x > > Don't Forget -17 < 5 - 2x < 29 - 5 - 5 - 5
Interval Notation
-22 < -2x < 24
>
>
Don't Forget
11 > x > -12
-12 11
Jeff Bivin -- LZHS

6. Solve for x Alternative NO YES NO -17 < 5 - 2x < 29 -17 = 5 - 2x
Interval Notation
-17 = 5 - 2x
5 -2 x = 29
-22 = -2x
-2x = 24
11 = x
x = -12
-12 11 NO
Try -100.
Does it work?
YES NO
Try 100.
Does it work?
Try 0.
Does it work?
Jeff Bivin -- LZHS

7. Solve for x Alternative NO YES NO 11 < 5 + 2x < 19 11 = 5 + 2x
Interval Notation
11 = 5 + 2x
5 + 2 x = 19
6 = 2x
2x = 14
3 = x
x = 7
11 < 5 + 2(0) < 19 3 7 NO
11 < 5 + 2(5) < 19
Try 0.
Does it work?
YES NO
Try 10.
Does it work?
Try 5.
Does it work?
11 < 5 + 2(10) < 19
Jeff Bivin -- LZHS

8. | 2x + 3 | < 9 NO YES NO 2x + 3 = 9 2x + 3 = -9 2x = 6 2x = -12
Interval Notation
2x + 3 = 9
2x + 3 = -9
2x = 6
2x = -12
x = 3
x = -6
| 2(-10) + 3 | < 9 -6 3 NO
Try -10.
Does it work?
YES NO
| 2(0) + 3 | < 9
Try 100.
Does it work?
Try 0.
Does it work?
| 2(100) + 3 | < 9
Jeff Bivin -- LZHS

9. | 2x - 7 | > 11 YES NO YES 2x - 7 = 11 2x - 7 = -11 2x = 18 2x = -4
Interval Notation
2x - 7 = 11
2x - 7 = -11
2x = 18
2x = -4
x = 9
x = -2
| 2(-10) - 7 | > 11 -2 9
YES
Try -10.
Does it work? NO
YES
| 2(0) - 7 | > 11
Try 10.
Does it work?
Try 0.
Does it work?
| 2(10) - 7 | > 11
Jeff Bivin -- LZHS

10. Absolute value is always positive
| 5x + 2 | > -4
Absolute value is always positive
Isn’t a positive number always bigger than a negative number?
YES!
Therefore, any number will work.
Interval Notation
Jeff Bivin -- LZHS

11. 1 < | x - 2 | < 5 -3 1 3 7 Interval Notation: | x - 2 | = 1
Try
-10
Try
Try 2
Try 5
Try 10
1 < | | < 5
1 < | | < 5
1 < | | < 5
1 < | | < 5
1 < | | < 5
Interval Notation:
1 < | -12 | < 5
1 < | -2 | < 5
1 < | 0 | < 5
1 < | 3 | < 5
1 < | 8 | < 5
1 < 12 < 5
1 < 2 < 5
1 < 0 < 5
1 < 3 < 5
1 < 8 < 5
Jeff Bivin -- LZHS

12. 5 < | x + 9 | < 11 2 -20 -14 -4 Interval Notation: | x + 9 | = 5
Try
-30
Try
-15
Try
-10
Try
Try 10
5 < | | < 11
5 < | | < 11
5 < | | < 11
5 < | | < 11
5 < | | < 11
Interval Notation:
5 < | -21 | < 11
5 < | -6 | < 11
5 < | -1 | < 11
5 < | 9 | < 11
5 < |19 | < 11
5 < 21 < 11
5 < 6 < 11
5 < 1 < 11
5 < 9 < 11
1 < 19 < 5
Jeff Bivin -- LZHS

13. -2 < | x + 1 | < 5 -6 4 Interval Notation: | x + 1 | = 5
A negative is always less than an absolute value!
x + 1 = 5
x + 1 = -5
Therefore, we have NO partitions here.
x = 4
x = -6 -6 4
Try
-10
Try
Try 10
-2 < | | < 5
-2 < | | < 5
-2 < | | < 5
Interval Notation:
-2 < | -9 | < 5
-2 < | 1 | < 5
-2 < | 11 | < 5
-2 < 9 < 5
-2 < 1 < 5
-2 < 11 < 5
Jeff Bivin -- LZHS

14. 7 < | 2x + 11 | < 3 ?
What number is bigger than 7 and smaller than 3 at the same time?
There isn’t any! Ø
Jeff Bivin -- LZHS

15. (x + 3)(x – 4)(x + 1)(x – 2) > 0
Try
-10 -3
Try -2 -1
Try 2
Try 3 4
Try 10
(x + 3) (x – 4) (x + 1) (x – 2) > 0
x = -10: ( – ) ( – ) ( – ) ( – ) = + > 0 ? YES
x = -2: ( + ) ( – ) ( – ) ( – ) = – > 0 ? NO
Interval Notation:
x = 0: ( + ) ( – ) ( + ) ( – ) = + > 0 ? YES
x = 3: ( + ) ( – ) ( + ) ( + ) = – > 0 ? NO
x = 10: ( + ) ( + ) ( + ) ( + ) = + > 0 ? YES
Jeff Bivin -- LZHS