1. Warm Up Solve |x – 5| = 4 x – 5 = 4 or x – 5 = -4 +5 +5 +5 +5
x = or x = 1
2x – 7 = or 2x – 7 = -5
2x = x = 2
x = or x = 1
There is no solution.

2. Inequalities and Absolute Value
Find points on the number line below that satisfy the inequality |x|  5 -6 -4 -2 2 4 6 8 -8
The inequality is true for any number between –5 and 5.
We write the solution as the conjunction x  -5 and x  5.
The solution set is {x | -5  x  5} -6 -4 -2 2 4 6 8 -8

3. Inequalities and Absolute Value
To solve an inequality of the form |A| < b where b is positive
we solve the conjunction -b < A < b .
Example 1: Solve |5x|  15
Solve the conjunction -15  5x  15 by rewriting it using and:
-15  5x and 5x  15
-3  x and x  3
Rewrite with the shortcut for conjunctions and graph:
-3  x  3 -6 -4 -2 2 4 6 8 -8

4. Inequalities and Absolute Value
Example 2: Solve |x + 5|  6
Rewrite: -6  x + 5  6
-6  x and x + 5  6
-11  x and x  1
Rewrite with the shortcut for conjunctions and graph:
-11  x  1
-10 -8 -6 -4 -2 2 4
-12

5. Inequalities and Absolute Value
Example 3: Solve |3x – 4| < 2
Rewrite: -2  3x – 4  2
-2 < 3x and x – 4 < 2
2 < 3x and x < 6
(2/3) < x and x < 2
Rewrite with the shortcut for conjunctions and graph:
(2/3) < x < 2 -1 1 2 3 -2

6. Inequalities and Absolute Value
Find points on the number line below that satisfy the inequality |x|  4 -6 -4 -2 2 4 6 8 -8
The inequality is true for any number larger than 4 and smaller than –4.
We write the solution as the disjunction x  4 or x  -4
The solution set is {x | x  -4 or x  4} -6 -4 -2 2 4 6 8 -8

7. Inequalities and Absolute Value
To solve an inequality of the form |A| > b where b is positive
we solve the disjunction A < -b or A > b .
Example 4: Solve |4x|  12
Solve the disjunction:
4x  or 4x  12
x  or x  3 -6 -4 -2 2 4 6 8 -8

8. Inequalities and Absolute Value
Example 5: Solve |x – 4| > 4
Solve the disjunction:
x – 4 < or x – 4 > 4
x < or x > 8 -6 -4 -2 2 4 6 8 -8

9. Inequalities and Absolute Value
Example 6: Solve |2x + 4|  12
Solve the disjunction:
2x + 4  or 2x + 4  12
2x  or 2x  8
x  or x  4 -6 -4 -2 2 4 6 8 -8