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5.7 Quadratic Inequalities

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You have already solved linear inequalities in two variables by graphing. You can use a similar procedure to graph quadratic inequalities. y ax 2 + bx + c

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Graph y x 2 – 7x + 10. Step 1 Graph the boundary of the related parabola y = x 2 – 7x + 10 with a solid curve. Its y-intercept is 10, its vertex is (3.5, –2.25), and its x-intercepts are 2 and 5.

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Step 2 Shade above the parabola because the solution consists of y-values greater than those on the parabola for corresponding x-values.

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Graph the inequality. y 2x 2 – 5x – 2

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Graph each inequality. Whiteboards y < –3x 2 – 6x – 7

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Quadratic inequalities in one variable, such as ax 2 + bx + c > 0 (a 0), have solutions in one variable that are graphed on a number line. For and statements, both of the conditions must be true. For or statements, at least one of the conditions must be true. Reading Math

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Solve the inequality by using tables or graphs. You now need a calculator (graphing, of course) x 2 + 8x + 20 5 Use a graphing calculator to graph each side of the inequality. Set Y 1 equal to x 2 + 8x + 20 and Y 2 equal to 5. Identify the values of x for which Y 1 Y 2.

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Continued The parabola is at or above the line when x is less than or equal to –5 or greater than or equal to –3. So, the solution set is x –5 or x –3 or (–, –5] U [–3, ). The table supports your answer. –6 –4 –2 0 2 4 6 The number line shows the solution set.

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Solve the inequality by using tables and graph. Now you try x 2 + 8x + 20 < 5 The solution set is –5 < x < –3 or (–5, –3). –6 –4 –2 0 2 4 6

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The number lines showing the solution sets in the last example are divided into three distinct regions by the points –5 and –3. These points are called critical values. By finding the critical values, you can solve quadratic inequalities algebraically.

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Solve the inequality by using tables and graph. 2x 2 – 5x + 1 1 Whiteboards The solution set is (–, 0] or [2.5, ) –6 –4 –2 0 2 4 6

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Solve the inequality x 2 – 10x + 18 –3 by using algebra. Solving Quadratic Inequalities algebraically Step 1 Write the related equation. x 2 – 10x + 18 = –3

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Example 3 Continued Write in standard form. Step 2 Solve the equation for x to find the critical values. x 2 –10x + 21 = 0 x – 3 = 0 or x – 7 = 0 (x – 3)(x – 7) = 0 Factor. Zero Product Property. Solve for x. x = 3 or x = 7 The critical values are 3 and 7. The critical values divide the number line into three intervals: x 3, 3 x 7, x 7.

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