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Aim: Rational Inequalities Course: Adv. Alg. & Trig. Aim: How do we solve various inequality problems? Do Now: Graph the solution set of 3x + 5 > 14

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Aim: Rational Inequalities Course: Adv. Alg. & Trig. Rational Inequalities What values of x makes this statement true? cross-multiply? distribute isolate variable divide check: x = 5 also true!

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Aim: Rational Inequalities Course: Adv. Alg. & Trig. Rational Inequalities What values of x makes this statement true? standard form to find zeros subtract - LCD Combine & Simplify Find zeros of Numerator & Denominator 4x – 12 = 0 4 – x = 0 x = 3 x = 4 critical values

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Aim: Rational Inequalities Course: Adv. Alg. & Trig. Rational Inequalities - Algebraically Test Interval Representative x-value Value of Polynomial (- , 3) x = 0 y = -3 (3, 4) x = 3.5 (4, ) x = 10 y = 4 y = -4 2/3 Find zeros of Numerator & Denominator 4x – 12 = 0 4 – x = 0 x = 3 x = 4 critical values ( )

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Aim: Rational Inequalities Course: Adv. Alg. & Trig. Rational Inequalities - Algebraically Algebraically, find the zeros, the critical numbers and test intervals of Standard form to find zeroes Add fractionsSimplify to find zeroes critical numbers: x = 8; x = 5 Note: rational expressions change sign at its zeros and its undefined values. =

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Aim: Rational Inequalities Course: Adv. Alg. & Trig. Rational Inequalities - Algebraically Algebraically find the zeros, the critical numbers and test intervals of critical numbers: x = 8 x = 5 Test Interval Representative x-value Value of Polynomial (- , 5) x = 0 y = -8/5 (5, 8] x = 6 [8, ) x = 10 y = 2 y = -2/5

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Aim: Rational Inequalities Course: Adv. Alg. & Trig. Rational Inequalities Solve Standard form Graphically: y is negative from (- , 5) and [8, ) (8, 0) < 0

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Aim: Rational Inequalities Course: Adv. Alg. & Trig. Using the Graphing Calculator = (-x + 8)/(x – 5) Y1Y1 2ndMATH 6 GRAPH 0 ≤ 0

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Aim: Rational Inequalities Course: Adv. Alg. & Trig. Model Problem Solve Test Interval Representative t-value Value of Polynomial (- , 0) (0, 1.25] [1.25, ) t = -1 t = 1 t = 10 f(t) = -9/4 f(t) = 1/4 f(t) = -36/40 Note: rational expressions change sign at its zeros and its undefined values. = = critical numbers: t = 0; t = 1.25 (1.25, 0)(0, 0)

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