1. Solving Compound and Absolute Value Inequalities
Chapter 1.6
By Courtney McCall

2. Vocabulary
Interval Notation- a way to describe the solution set of an inequality.
Infinity- without bound, or continues without end.
Absolute Value- a number’s distance from 0 on the number line.
3 units 3 units
Notice that the graph of |x|<3 is the same as the graph of x>-3 and x<3.

3. Vocabulary
Compound Inequality- consists of two inequalities joined by the word AND or the word OR.
Intersection- the graph of a compound inequality containing AND.
Union- the graph of a compound inequality containing OR.

4. Extra Facts must switch sign when dividing by a negative number
> or < = open circle
≥ or ≤ = closed circle
|x|<-5 = empty set
|x|> -5 = infinite solutions
within= ≥ or ≤
between= > or <

5. Compound Inequalities

6. Solving “And” Compound Inequalities
solve separately or solve both sides together
“and” compound inequalities are true if both inequalities are true
8<3x-7≤ 23
8<3x-7 and 3x-7≤23
15<3x 3x≤30
5<y and y≤10 5<x≤10
8<3x-7≤23
15<3x≤30
5<x≤10 or

7. Example & Practice Problems (AND)
1. -7≤4x-3≤ >6x+4>16
-4≤4x≤2
-1≤x≤ >x>2
when you divide

8. Solving “Or” Compound Inequalities
solve separately
“or” compound inequalities are true if one or more inequalities are true
x+6<-4 or 3x≥14
x+6<-4 or 3x≥14
x<-10 x≥14/3

9. Example & Practice Problems (OR)
1. -4x+2<-26 or 6x-3< x-7≥-12 or -3x+2≥38
-4x< or 6x<-24
x>7 or x< x≥-5 or x≤-12

10. Absolute Value Inequalities

11. Solve Absolute Value Inequalities
|x|>a = greatOR than |x|<a =less thAND
|x|≥a = greatOR than or equal to |x|≤a =less thAND or equal to
make 2 equations
original
switched sign w/ negative
|4x+5|>7
4x+5>7 or 4x+5<-7
4x> x<-12
x>½ x<-3
x>½ or x<-3
|4x+5|<7
4x+5< and 4x+5>-7
4x< x>-12
x<½ x>-3
½>x>-3

12. Example & Practice Problems
1. |-4x|> |6x|≤12
-4x>16 or -4x<-16
x<-4 or x>4
2≥x≥-2