Solving Compound and Absolute Value Inequalities Chapter 1.6 By Courtney McCall

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. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 3 units 3 units Notice that the graph of |x|<3 is the same as the graph of x>-3 and x<3. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5

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. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5

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 <

Compound Inequalities

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

Example & Practice Problems (AND) 1. -7≤4x-3≤-1 2. 28>6x+4>16 -4≤4x≤2 -1≤x≤0.5 4>x>2 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 when you divide

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

Example & Practice Problems (OR) 1. -4x+2<-26 or 6x-3<-27 2. x-7≥-12 or -3x+2≥38 -4x<-28 or 6x<-24 x>7 or x<-4 x≥-5 or x≤-12 -10-8 -6 -4 -2 0 2 4 6 8 10 12 14 16 -13-12-11-10-9-8 -7 -6 -5 -4 -3 -2 -1 0

Absolute Value Inequalities

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>2 4x<-12 x>½ x<-3 x>½ or x<-3 |4x+5|<7 4x+5<7 and 4x+5>-7 4x<2 4x>-12 x<½ x>-3 ½>x>-3

Example & Practice Problems 1. |-4x|>16 2. |6x|≤12 -4x>16 or -4x<-16 x<-4 or x>4 2≥x≥-2 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5