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2.6 Solving Linear Inequalities 1. Represent solutions to inequalities graphically and using set notation. 2. Solve linear inequalities. Inequalities < is less than > is greater than is less than or equal to is greater than or equal to Inequality always points to the smaller number. True or False? 44 4>4 True False Represent inequalities: x > 4 is the same as {5, 6, 7…} False Graphically Interval Notation Set-builder Notation Graphing Inequalities Parentheses/bracket method : If the variable is on the left, the arrow points the same direction as the inequality. Parentheses: endpoint is not included <, > Bracket: endpoint is included ≤, ≥ x<2 x≥2 Open Circle/closed circle method: Open Circle: endpoint is not included <, > Closed Circle: endpoint is included ≤, ≥ x<2 x≥2 Inequalities – Interval Notation [( smallest, largest )] Parentheses: endpoint is not included <, > Bracket: endpoint is included ≤, ≥ Infinity: always uses a parenthesis x<2 ( –∞, 2) x≥2 [2, ∞) 4<x<9 3-part inequality (4, 9) Inequalities – Set-builder Notation {variable | condition } pipe { x | x 5} The set of all x such that x is greater than or equal to 5. x<2 ( –∞, 2) {x|x<2} [2, ∞) { x | x ≥ 2} (4, 9) { x | 4 < x < 9} x≥2 4<x<9 Inequalities Graph, then write interval notation and set-builder notation. x≥5 [ Interval Notation: [ 5, ∞) Set-builder Notation: { x | x ≥ 5} x < –3 ) Interval Notation: (– ∞, –3) Set-builder Notation: { x | x < –3 } Inequalities Graph, then write interval notation and set-builder notation. 1<a<6 ( ) Interval Notation: ( 1, 6 ) Set-builder Notation: { a | 1 < a < 6 } –7 < x ≤ 3 ( ] Interval Notation: (– 7, –3] Set-builder Notation: { x | –7 < x ≤ 3 } Inequalities 4<5 4<5 4+1<5+1 4–1<5–1 5<6 3<4 True True The Addition Principle of Inequality If a < b, then a + c < b + c for all real numbers a, b, and c. Also true for >, , or . Inequalities 4<5 4 (2) < 5 (2) 8 < 10 True 4<5 4 (–2) < 5 (–2) –8 < –10 –8 > –10 False If we multiply (or divide) by a negative, reverse the direction of the inequality!!!!! The Multiplication Principle of Inequality If a < b, then ac < bc if c is a positive real number. If a < b, then ac > bc if c is a negative real number. The principle also holds true for >, , and . Solving Inequalities If we multiply (or divide) by a negative, reverse the direction of the inequality!!!!! 4 x 16 4 4 x 4 4 x 16 4 4 x 4 4 x 16 4 4 x 4 Solving Inequalities Solve then graph the solution and write it in interval notation and set-builder notation. 3x 4 7 4 4 Don’t write = ! 3x 3 3 ( 3 x 1 Interval Notation: ( 1, ∞ ) Set-builder Notation: { x | x > 1 } Solving Inequalities Solve then graph the solution and write it in interval notation and set-builder notation. 4 9 k 4 k 19 4k 4k 4 5 k 19 4 4 5 k 15 5 5 ] k 3 Interval Notation: (– ∞, –3 ] Set-builder Notation: { k | k ≤ –3 } Solving Inequalities Solve then graph the solution and write it in interval notation and set-builder notation. 5 3 p 10 5 3 p 3 10 3 5 p 30 5 5 ) p 6 Interval Notation: (– ∞, 6 ) Set-builder Notation: { p | p < 6 } Solving Inequalities Solve then graph the solution and write it in interval notation Moving variable to the right. and set-builder notation. 1 5 6 m 7 1 2 12 m 14 15 m 5 3 m 1 12 m 1 1 10 6 m 7 10 3 m 1 5 2 2 6 m 7 5 3 m 1 12 m 14 15 m 5 15 m 14 3 m 5 5 5 9 3m 3 3 3 m 15 m 3 m 14 5 14 3m 9 3 12 m 3 m 3 [ 14 Interval Notation: [– 3, ∞ ) Set-builder Notation: { m | m ≥ – 3 }