1. Chapter 1.6 Absolute Value Equations and inequalities
How can I write and solve absolute-value equations and inequalities?
Entry Task: There is a tour group recruiting members. The group has a budget between 1400 dollars and 2000 dollars. Each tourist needs to pay 150 dollars for the trip, and there is a fixed fee of 50 dollars for the tour bus. Write the inequality that describes how many members they can recruit within the budget? Solve.

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Homework Check #4

3. 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.

4. 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.

5. Example 2A: Solving Absolute-Value Equations
Solve the equation.
|–3 + k| = 10
–3 + k = 10 or –3 + k = –10
k = 13 or k = –7
Solve the equation.
|6x| – 8 = 22
|6x| = 30
6x = 30 or 6x = –30
x = 5 or x = –5

6. Example 2B: 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

7. The solutions of |x| < 3 are the points that are less than 3 units from zero. The solution is an “and” statement: –3 < x < 3.
The solutions of |x| > 3 are the points that are more than 3 units from zero. The solution is an “or” statement:
x < –3 or x > 3.

8. Example 3A: Solving Absolute-Value Inequalities
Solve the inequality. Then graph the solution.
|–4q + 2| ≥ 10
Rewrite the absolute value as or.
–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
–3 –2 –

9. Example 3B: Solving Absolute-Value Inequalities
Solve the inequality. Then graph the solution.
|0.5r| – 3 ≥ –3
Isolate the absolute value as or.
|0.5r| ≥ 0
Rewrite the absolute value as or.
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 –
(–∞, ∞)

10. Solve the compound inequality. Check for Extraneous Solutions
Check It Out! Example 4b
Solve the compound inequality. Check for Extraneous Solutions
|3x +2| = 4x + 5
x = –3 or x =-1

11. Assignment #5
Pg 46 #12-24 x 3, x3 (9 pts)

12. Solve the compound inequality.
Check It Out! Example 4b
Solve the compound inequality.
–2|x +5| > 10
Divide both sides by –2, and reverse the inequality symbol.
|x + 5| < –5
Rewrite the absolute value as a conjunction.
x + 5 < – x + 5 > 5
Subtract 5 from both sides of each inequality.
x < –10 or x > 0

13. Solve. Then graph the solution.
Lesson Quiz: Part I
Solve. Then graph the solution. 1.
y – 4 ≤ –6 or 2y >8
{y|y ≤ –2 ≤ or y > 4}
–4 –3 –2 – 2.
–7x < 21 and x + 7 ≤ 6
{x|–3 < x ≤ –1}
–4 –3 –2 –
Solve each equation.
3. |2v + 5| = 9
4. |5b| – 7 = 13
2 or –7
+ 4

14. Solve. Then graph the solution.
Lesson Quiz: Part II
Solve. Then graph the solution. 5.
|1 – 2x| > 7
{x|x < –3 or x > 4}
–4 –3 –2 – 6.
|3k| + 11 > 8 R
–4 –3 –2 – 7.
–2|u + 7| ≥ 16 ø

15. Do Now: What’s the ERROR?1. 2.