Solving Linear Inequalities 2.6 Solving Linear Inequalities 1. Represent solutions to inequalities graphically and using set notation. 2. Solve linear inequalities.

Inequalities Inequality always points to the smaller number. < 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? 4  4 4 > 4 x > 4 is the same as {5, 6, 7…} True False False Represent inequalities: Graphically Interval Notation Set-builder Notation

Graphing Inequalities If the variable is on the left, the arrow points the same direction as the inequality. Parentheses/bracket method : 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)

The set of all x such that x is greater than or equal to 5. 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 x < 2 ( –∞, 2) { x | } x ≥ 2 [2, ∞) { x | x ≥ 2} 4 < x < 9 (4, 9) { x | 4 < x < 9}

Inequalities x ≥ 5 x < –3 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 1 < a < 6 –7 < x ≤ 3 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 The Addition Principle of Inequality 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 < 5 4 (–2) < 5 (–2) 4 (2) < 5 (2) –8 < –10 –8 > –10 8 < 10 True 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!!!!!

Solving Inequalities Solve then graph the solution and write it in interval notation and set-builder notation. Don’t write = ! ( 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. ] 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. ) Interval Notation: (– ∞, 6 ) Set-builder Notation: { p | p < 6 }

Solving Inequalities Solve then graph the solution and write it in interval notation and set-builder notation. Moving variable to the right. [ Interval Notation: [– 3, ∞ ) Set-builder Notation: { m | m ≥ – 3 }