Mastering Linear Inequalities: Graphical Representations and Solution Methods

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

2. Inequalities 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

3. 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

4. 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)

5. Inequalities – Set-builder Notation {variable | condition } pipe { x|x  5} The set of all xsuch thatx is greater than or equal to 5. x < 2 x < 2 { x | } ( –∞, 2) x ≥ 2 [2, ∞) { x | x ≥ 2} 4 < x < 9 (4, 9) { x | 4 < x < 9}

6. 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 }

7. ( ] ( ) 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 }

8. 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 .

9. Inequalities 4 < 5 4 < 5 4 (–2) < 5 (–2) 4 (2) < 5 (2) –8 < –10 –8 > –10 8 < 10 False True 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 .

10. Solving Inequalities If we multiply (or divide) by a negative, reverse the direction of the inequality!!!!!

11. 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 }

12. 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 }

13. 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 }

14. 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 }