8.2 Linear Inequalities

We will remind ourselves how to solve inequalities and graph on a number line as well as the coordinate plane Inequality a < x < b a ≤ x ≤ b a ≤ x < b a < x ≤ b x > a x ≥ a x < a x ≤ a Interval (a, b) [a, b] [a, b) (a, b] (a, ∞) [a, ∞) (–∞, a) (–∞, a] Graph a b a b a b a b a a a a

In a compound inequality, two conditions are given - conjunction - disjunction  “and” 6 – 10x < 5 – 10x < –1 b)  the intersection of the sets  “or”  the union of the sets Ex 1) Solve and graph on a number line. Express in interval notation. a) 3 ≤ 4x – 1 < 7 3 ≤ 4x – 1 4 ≤ 4x 1 ≤ x x ≥ 1 *deal with one part at a time* 4x – 1 < 7 4x < 8 x < 2 [1, 2) 1 2

Ex 1) Solve and graph on a number line. Express in interval notation. c) Absolute Value: *Remember could mean x = 2 or x = –2 *Hint: ││< # say “less thAND” 2x – 3 ≥ 2 or 2x – 3 < –4 2x ≥ 5 2x < –1 say “greatOR” ││> # Ex 2) Solve and graph on a number line. Express in interval notation. a) │6 – 4x│ ≤ 2 “and” 6 – 4x ≤ 2 –4x ≤ –4 x ≥ 1 and 6 – 4x ≥ –2 –4x ≥ –8 x ≤ 2 [1, 2] 1 2

Ex 2) Solve and graph on a number line. Express in interval notation. b) │2x + 1│ > 3 “or” 2x + 1 > 3 2x > 2 x > 1 or 2x + 1 < –3 2x < –4 x < –2 1 –2 (–∞, –2)  (1, ∞) Graphing linear inequalities in two variables in the coordinate plane dotted line ≤ or ≥ solid line y < or y ≤ shade “below” y > or y ≥ shade “above” The two regions the coordinate plane is divided into is called half-plane. The line is the boundary. closed half-plane: solid line open half-plane: dotted line

Ex 3) Graph in the coordinate plane a)3x – y > 4 –y > –3x + 4 y < 3x – 4 shade below b) –3 ≤ 2x + y < 6 –3 ≤ 2x + y –y ≤ 2x + 3 y ≥ –2x – 3 shade above do each line 2x + y < 6 y < –2x + 6 shade below

c)y < –│x + 2│ abs value reflect over x-axis shifted 2 left dotted line shade below

Homework #802 Pg 398 #1, 5, 9, 13, 16, 21, 24, 29, 34, 38, 39, 43, 44