1. Solving Absolute Value Equations and Inequalities
Section 2.8
Solving Absolute Value Equations and Inequalities

2. Homework
Pg #20-27, 32-35, 59, 60

3. A compound statement is made up of more than one equation or inequality.
A disjunction is a compound statement that uses the word or.
Disjunction: x ≤ –3 OR x > 2 Set builder notation: {x|x ≤ –3 U x > 2}
A disjunction is true if and only if at least one of its parts is true.

4. A conjunction is a compound statement that uses the word and.
Conjunction: x ≥ –3 AND x < 2 Set builder notation: {x|x ≥ –3 x < 2}.
A conjunction is true if and only if all of its parts are true. Conjunctions can be written as a single statement as shown.
x ≥ –3 and x< –3 ≤ x < 2 U

5. Solving Compound Inequalities
Solve the compound inequality. Then graph the solution set.
6y < –24 OR y +5 ≥ 3
Solve both inequalities for y.
6y < –24
y + 5 ≥3 or
y < –4
y ≥ –2
The solution set is all points that satisfy {y|y < –4 or y ≥ –2}.
–6 –5 –4 –3 –2 –
(–∞, –4) U [–2, ∞)

6. Solving Compound Inequalities
Solve the compound inequality. Then graph the solution set.
Solve both inequalities for c.
and
2c + 1 < 1
c ≥ –4
c < 0
The solution set is the set of points that satisfy both c ≥ –4 and c < 0.
–6 –5 –4 –3 –2 –
[–4, 0)

7. Recall that the absolute value of a number x, written |x|, is the distance from x to zero on the number line. Because absolute value represents distance without regard to direction, the absolute value of any real number is nonnegative.

8. Absolute-value equations and inequalities can be represented by compound statements. Consider the equation |x| = 3.
The solutions of |x| = 3 are the two points that are 3 units from zero. The solution is a disjunction: x = –3 or x = 3.

9. The solutions of |x| < 3 are the points that are less than 3 units from zero. The solution is a conjunction: –3 < x < 3.

10. The solutions of |x| > 3 are the points that are more than 3 units from zero. The solution is a disjunction:
x < –3 or x > 3.

11. Note: The symbol ≤ can replace <, and the rules still apply
Note: The symbol ≤ can replace <, and the rules still apply. The symbol ≥ can replace >, and the rules still apply.

12. Solving Absolute-Value Equations
Solve the equation.
This can be read as “the distance from k to –3 is 10.”
|–3 + k| = 10
Rewrite the absolute value as a disjunction.
–3 + k = 10 or –3 + k = –10
Add 3 to both sides of each equation.
k = 13 or k = –7

13. Solving Absolute-Value Equations
Solve the equation.
Isolate the absolute-value expression.
Rewrite the absolute value as a disjunction.
Multiply both sides of each equation
by 4.
x = 16 or x = –16

14. Solving Absolute-Value Inequalities with Disjunctions
Solve the inequality. Then graph the solution.
|–4q + 2| ≥ 10
Rewrite the absolute value as a disjunction.
–4q + 2 ≥ 10 or –4q + 2 ≤ –10
Subtract 2 from both sides of each inequality.
–4q ≥ 8 or –4q ≤ –12
Divide both sides of each inequality by –4 and reverse the inequality symbols.
q ≤ –2 or q ≥ 3

15. Solving Absolute-Value Inequalities with Disjunctions
Solve the inequality. Then graph the solution.
|0.5r| – 3 ≥ –3
Isolate the absolute value as a disjunction.
|0.5r| ≥ 0
Rewrite the absolute value as a disjunction.
0.5r ≥ 0 or 0.5r ≤ 0
Divide both sides of each inequality by 0.5.
r ≤ 0 or r ≥ 0
The solution is all real numbers, R.
–3 –2 –
(–∞, ∞)

16. Solving Absolute-Value Inequalities with Conjunctions
Solve the compound inequality. Then graph the solution set.
|2x +7| ≤ 3
Multiply both sides by 3.
Rewrite the absolute value as a conjunction.
2x + 7 ≤ 3 and 2x + 7 ≥ –3
Subtract 7 from both sides of each inequality.
2x ≤ –4 and 2x ≥ –10
Divide both sides of each inequality by 2.
x ≤ –2 and x ≥ –5

17. Solving Absolute-Value Inequalities with Conjunctions
Solve the compound inequality. Then graph the solution set.
Multiply both sides by –2, and reverse the inequality symbol.
|p – 2| ≤ –6
Rewrite the absolute value as a conjunction.
|p – 2| ≤ –6 and p – 2 ≥ 6
Add 2 to both sides of each inequality.
p ≤ –4 and p ≥ 8
Because no real number satisfies both p ≤ –4 and p ≥ 8, there is no solution. The solution set is ø.