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Mastering Linear Inequalities: Solve, Graph & Interpret with Precision

Learn to solve and graph inequalities on a number line and coordinate plane, understand interval notation, and tackle compound inequalities with ease. Enhance your skills in linear inequalities and grasp the concepts of conjunction and disjunction. Practice solving absolute value inequalities, and advance your graphing expertise in two variables on the coordinate plane. Homework exercises included.

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Mastering Linear Inequalities: Solve, Graph & Interpret with Precision

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  1. 8.2 Linear Inequalities

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

  3. In a compound inequality, two conditions are given - conjunction - disjunction  “and”  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) • 6 – 10x < 5 • – 10x < –1 • 3 ≤ 4x – 1 < 7 • 3 ≤ 4x – 1 • 4 ≤ 4x • 1 ≤ x • x ≥ 1 *deal with one part at a time* b) • 4x – 1 < 7 • 4x < 8 • x < 2 2 1 [1, 2)

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

  5. Ex 2) Solve and graph on a number line. Express in interval notation. b) │2x + 1│ > 3 “or” or 2x + 1 < –3 2x < –4 x < –2 2x + 1 > 3 2x > 2 x > 1 –2 1 (–∞, –2)  (1, ∞) Graphing linear inequalities in two variables in the coordinate plane < or > 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

  6. Ex 3) Graph in the coordinate plane 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

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

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

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