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Solving Nonlinear Inequalities Digital Lesson

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Quadratic Inequality 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. Example: x 2 – 3x is a quadratic inequality since it can be written 1x 2 + (– 3)x Example: 3x 2 < x + 5 is a quadratic inequality since it can be written 3x 2 + (–1)x + (– 5) < 0. Example: x 2 + 3x x is not a quadratic inequality since it is equivalent to 3x 4 0. for a, b, c real numbers.

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Solution 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 + 3x 4 0 ? x x 2 + 3x – 4 x 2 + 3x – 4 0 Solution? 1( 1) 2 + 3( 1) – true yes 0(0) 2 + 3(0) – true yes 0.5(0.5) 2 + 3(0.5) – true yes 1(1) 2 + 3(1) – 4 0 true yes 2(2) 2 + 3(2) – false no 3(3) 2 + 3(3) – false no

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Solution Set The solution set of an inequality is the set of all solutions. Study the graph of the solution set of x 2 + 3x 4 0. The solution set is {x | 4 x 1}. The values of x for which equality holds are part of the solution set. These values can be found by solving the quadratic equation associated with the inequality. x 2 + 3x 4 = 0 Solve the associated equation. (x + 4)(x 1) = 0 Factor the trinomial. x = 4 or x = 1 Solutions of the equation ][

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Solving a Quadratic Inequality To solve a quadratic inequality: 1. If necessary, rewrite the quadratic inequality so that zero appears on the right, then factor. 2. On the real number line, draw a vertical line at the numbers that make each factor equal to zero. 3. For each factor, place plus signs above the number line in the regions where the factor is positive, and minus signs where the factor is negative. 5. Express the solution set using set-builder notation and a graph on a real number line. 4. Observe the sign of the product of the factors for each region, to determine which regions will belong to the solution set.

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Product is positive. Product is negative. Product is positive. The product of the factors is negative. Example: Solve and Graph the Solution Set Example: Solve and graph the solution set of x 2 6x + 5 < 0. (x 1)(x 5) < 0 Factor. x = 1 Solve for each factor equal to zero. {x | 1 < x < 5} Solution set in set-builder notation x – 1 x – 5 Factors Draw vertical lines indicating the numbers where each factor equals zero. For each region, identify if each factor is positive or negative. x 1 = 0x 5 = 0 x = 5 )( Draw the solution set. Rounded parentheses indicate a strict inequality. – – – – – –– – – – – + + +

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 The product of the factors is positive. Example: Solve and Graph the Solution Set Example: Solve and graph the solution set of x 2 x 6. (x + 2)(x 3) 0 Factor. x = 2, 3 Numbers where each factor equals zero. {x | x 2 or x 3} Solution set in set-builder notation [] – – – – – – – – – – – + x + 2 x – 3 Draw vertical lines where each factor equals zero. Indicate positive and negative regions for each factor. Draw solution set. Square brackets are used since the inequality is. x 2 x 6 0 Rewrite the inequality so that zero appears on the right.

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Solve a Cubic Inequality Cubic inequalities can be solved similarly. Example: Solve and graph the solution set of x 3 + x 2 9x 9 > 0. x 2 (x + 1) 9(x + 1) > 0 Factor by grouping. (x 2 9)(x + 1) > 0 Numbers where each factor equals zero. (x + 3)(x 3)(x + 1) > 0 x = 3, +3, x + 3 x + 1 x – 3 + – – – – – – – + – {x | 3 3} Solution set Draw three vertical lines. Indicate positive and negative regions for each of the three factors. ) ( (

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Note that 2 will not be part of the solution set since the expression is not defined when the denominator is zero. Example: Solve an Inequality Involving a Rational Function Inequalities involving rational functions can be solved similarly. Example: Solve and graph the solution set of. Find the numbers for which each factor equals zero. {x | x 1 or x > 2} Solution set – – – – – – – – – – – – x + 1 x – 2 (x + 1) = 0 x = 1 (x 2) = 0 x = 2 ( ] There are two regions where the quotient of the two factors is positive.

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Expression is undefined at these points. The quotient is negative. Example: Solve and Graph the Solution Set Example: Solve and graph the solution set of. x + 2 = 0 (x 1)(x + 3) = 0 x = 1, 3 {x | x < 3 or 2 < x < 1} Solution set x = 2 Factor x 1 x + 3 x + 2 –– – – – –– – – – – – – ) ( )

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example: Word Problem Example: One leg of a right triangle is 2 inches longer than the other. How long should the shorter leg be to ensure that the area of the triangle is greater than or equal to 4? x = shorter leg x + 2 = other leg x x + 2 Area of triangle Solve: The shorter leg should be at least 2 inches long. Since length has to be positive, the answer is x – – – – – – x + 4 x – 2 []

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