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Solving Linear Inequalities `

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Warm-up -4 < x ≤ 6 x ≤ -4 or x>6 -9-9 -8-8 -7-7 -6-6 -5-5 -4-4 -3-3 -2-2 -1 0123456789 -9-9 -8-8 -7-7 -6-6 -5-5 -4-4 -3-3 -2-2 -1 0123456789 -1 < x ≤ 9 -9-9 -8-8 -7-7 -6-6 -5-5 -4-4 -3-3 -2-2 0123456789 x ≤ 1 or x>7 -9-9 -8-8 -7-7 -6-6 -5-5 -4-4 -3-3 -2-2 -1 0123456789

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Solving Compound Inequalities -8 < 2x ≤ 10 2 2 2 -4 < x ≤ 5 -9-8-7-6-5-4-3-20123456789 Example 1 This compound inequality is a conjunction because it uses the word “and”. x is less than –4 and greater than 5.

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2-step Compound Inequalities 2x-1 < -9 or 2x-1 ≥ 5 -9-8-7-6-5-4-3-20123456789 Example 2 2x-1 < -9 +1 2x < -8 22 x < -4 2x-1 ≥ 5 +1 2x ≥ 6 22 x ≥ 3 x > -4 or x ≥ 9 This compound inequality is a disjunction because it uses the word “or”.

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Your Turn 2x < -12 or 2x ≥ 10-11 < 2x+1 ≤ 7 3 < 3x < 9 2x-3 < -15 or 2x-3 ≥ 7 -13 ≤ 5x-2 < 18 4x < -32 or 4x ≥ 40 -6 < x ≤ 3 x<-6 or x ≥ 5 1 < x < 3 x<-6 or x ≥ 5 -11/5 ≤ x < 4x<-8 or x ≥ 10

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6 A quadratic inequality in one variable is an inequality which can be written in the form ax 2 + bx + c > 0 ( a 0) The symbols , , and may also be used. for a, b, c real numbers.

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A solution of a quadratic inequality in one variable is a number which, when substituted for the variable, results in a true inequality. Example : Which of the values of x are solutions of x 2 + 3 x − 4 0 ? − 1− 1( − 1) 2 + 3( − 1) – 4 − 6 0 true yes 0(0) 2 + 3(0) – 4 − 4 0 true yes 2(2) 2 + 3(2) – 4 6 0 false no 3(3) 2 + 3(3) – 4 14 0 false no x x 2 + 3 x – 4 x 2 + 3 x – 4 0 Solution?

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Find the solution set of x 2 + 3 x − 4 0. x 2 + 3 x − 4 0 (x + 4)(x - 1) 0 (x +4) 0 (x - 1) 0 x -4 x 1 x ≤ -4 and x ≤ 1 is a conjunction, therefore -4 x 1 -9-8-7-6-5-4-3-20123456789

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Solve and graph the solution set of x 2 − 6 x + 5 < 0. YOUR TURN! 1 < x < 5 -9-8-7-6-5-4-3-20123456789

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Example : Solve and graph the solution set of x 2 − x 6. x 2 - x − 6 0 (x + 2)(x - 3) 0 (x +2) 0 (x - 3) 0 x -2 x 3 This inequality is a disjunction, therefore x -2 or x 3 -9-8-7-6-5-4-3-20123456789

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Solve and graph the solution set of x 2 − 7 x + 12 > 0. YOUR TURN! x 4 -9-8-7-6-5-4-3-20123456789

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