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

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Presentation on theme: "Solving Linear Inequalities 2.6 1.Represent solutions to inequalities graphically and using set notation. 2.Solve linear inequalities."— Presentation transcript:

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

2 is greater than  is less than or equal to  is greater than or equal to Inequalities Inequality always points to the smaller number. Represent inequalities: Graphically Interval Notation Set-builder Notation True or False? 4  4 4 > 4 TrueFalse x > 4 is the same as {5, 6, 7…} False

3 Graphing Inequalities 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 If the variable is on the left, the arrow points the same direction as the inequality.

4 Inequalities – Interval Notation [( smallest, largest )] Parentheses: endpoint is not included Bracket: endpoint is included ≤, ≥ Infinity: always uses a parenthesis x < 2 x ≥ 2 ( –∞, 2) [2, ∞) 4 < x < 9 (4, 9) 3-part inequality

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

6 Inequalities [ Graph, then write interval notation and set-builder notation. x ≥ 5 Interval Notation: Set-builder Notation: [ 5, ∞) { x | x ≥ 5} x < –3 Interval Notation: Set-builder Notation: (– ∞, –3) { x | x < –3 } )

7 Inequalities Graph, then write interval notation and set-builder notation. 1 < a < 6 Interval Notation: Set-builder Notation: ( 1, 6 ) { a | 1 < a < 6 } –7 < x ≤ 3 Interval Notation: Set-builder Notation: (– 7, –3] { x | –7 < x ≤ 3 } ( ) ( ]

8 Inequalities 4 < < < 6 True 4 < 5 4 – 1 < 5 – 1 3 < 4 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 (2) < 5 (2) 8 < 10 True 4 < 5 4 (–2) < 5 (–2) –8 < –10 False –8 > –10 The Multiplication Principle of Inequality If a < b, then ac < bc if c is a positive real number. If a bc if c is a negative real number. The principle also holds true for >, , and . If we multiply (or divide) by a negative, reverse the direction of the inequality!!!!!

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. ( Interval Notation: Set-builder Notation: ( 1, ∞ ) { x | x > 1 } Don’t write = !

12 Solving Inequalities Solve then graph the solution and write it in interval notation and set-builder notation. ] Interval Notation: Set-builder Notation: (– ∞, –3 ] { k | k ≤ –3 }

13 Solving Inequalities Solve then graph the solution and write it in interval notation and set-builder notation. ) Interval Notation: Set-builder Notation: (– ∞, 6 ) { p | p < 6 }

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


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