1. Absolute-Value Equations and Inequalities
9.3
Absolute-Value Equations and Inequalities
Equations with Absolute Value
Inequalities with Absolute Value

2. Warm Up Solve and graph each compound inequality.
1. -3x – 4 < -16 and 2x - 5 < 15
2. 4x + 2 > 10 or -2x + 6 > 16
3. 5 < -3x+2 < 8
4. 2x < -2 and x + 3 > 6
5. 5x > 15 or 6x - 4 < 32

3. Definition of absolute value.
The absolute value of a number is its distance from 0. Since absolute value is distance it is never negative.
Definition of absolute value.

4. Find each absolute value.
|-3|
|0|
|4|

5. Example Solution a) |x| = 6; b) |x| = 0; c) |x| = –2
a) |x| = 6 means that the distance from x to 0 is 6. Only two numbers meet that requirement. Thus the solution set is {–6, 6}.
b) |x| = 0 means that the distance from x to 0 is 0. The only number that satisfies this is zero itself. Thus the solution set is {0}.
c) Since distance is always nonnegative, |x| = –2 has no solution. Thus the solution set is

6. The Absolute-Value Principle for Equations
This brings us to…
The Absolute-Value Principle
for Equations
If |x| = p, then x = p or x = -p.
Note: The equation |x| = 0 is equivalent to the equation x = 0 and the equation |x| = –p has no solution.

7. Example: a) |2x +1| = 5; b) |3 – 4x| = –10
Solution
a) We use the absolute-value principle, knowing that 2x + 1 must be either 5 or –5:
|x| = p
|2x +1| = 5
Substituting
2x +1 = –5 or 2x +1 = 5
2x = –6 or x = 4
x = –3 or x = 2
The solution set is {–3, 2}. The check is left for the student.

8. (continued) Solution b) |3 – 4x| = –10
The absolute-value principle reminds us that absolute value is always nonnegative The equation |3 – 4x| = –10 has no solution. The solution set is
To apply the absolute value principle we must make sure the absolute value expression is ISOLATED.

9. Given that f (x) = 3|x + 5| – 4, find all x for which f (x) = 11
Solution Since we
are looking for f(x) = 11, we substitute:
f(x) = 11
Replacing f (x) with 3|x + 5| − 4
3|x+5| – 4 = 11
3|x + 5| = 15
|x + 5| = 5
x + 5 = –5 or x + 5 = 5
x = –10 or x = 0
The solution set is {–10, 0}. The check is left for the student.

10. When more than one absolute value expression appears…
Sometimes an equation has two absolute-value expressions like |x+1| = |2x|. This means that x+1 and 2x are the same distance from zero.
Since they are the same distance from zero, they are the same number or they are opposites.
So solve x+1 = 2x and x+1 = -2x separately to get your answers. As always check by substitution.

11. You are not done until you solve each equation.
Solve |3x – 5| = |8 + 4x|.
3x – 5 = 8 + 4x
This equation sets both sides the same.
This equation sets them opposite. Notice parenthesis.
3x – 5 = –(8 + 4x) or
You are not done until you solve each equation.

13. The Absolute-Value Inequalities Principle
For any positive number p and any expression x:
|x| < p is equivalent to –p < x < p. (conjunction)
|x| > p is equivalent to x < -p or x > p. (disjunction)
Similar rules apply for less than or equal, greater than or equal.

14. So the solutions of |X| < p are those numbers that satisfy –p < X < p.
And the solutions of |X| > p are those numbers that satisfy X < –p or p < X. (
–p p ) ( )
–p p

15. Solution Solve |x| < 3. Then graph.
The solutions of |x| < 3 are all numbers whose distance from zero is less than 3. By substituting we see that numbers like –2, –1, –1/2, 0, 1/3, 1, and 2 are all solutions. The solution set is {x| –3 < x < 3}. In interval notation, the solution set is (–3, 3). The graph is as follows: ( )

16. In interval notation, the solution set is The graph is as follows:
The solutions of are all numbers whose distance from zero is at least 3 units. The solution set is
In interval notation, the solution set is
The graph is as follows: [ ]

17. Solve |3x + 7| < 8. Then graph.
Solution
The number 3x + 7 must be less than 8 units from 0.
|X| < p
Replacing X with 3x + 7 and p with 8
|3x + 7| < 8
–8 < 3x + 7 < 8
–15 < 3x < 1
−5 < x < 1/3
The solution set is {x|–5 < x < 1/3}. The graph is as follows:
– /3 ( )

18. Solve |3x + 7| > 8. Then graph. Solution
The number 3x + 7 must be greater than 8 units from 0.
|X| < p
Replacing X with 3x + 7 and p with 8
|3x + 7| < 8
–8 < 3x + 7 < 8
–15 < 3x < 1
−5 < x < 1/3
The solution set is {x|–5 < x < 1/3}. The graph is as follows:
– /3 ( )