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Adapted from Walch Education

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Quadratic inequalities can be written in the form ax 2 + bx + c < 0, ax 2 + bx + c 0, ax 2 + bx + c > 0, or ax 2 + bx + c 0. The solutions to quadratic inequalities are written as intervals. An interval is the set of all real numbers between two given numbers. The two numbers on the ends are the endpoints. The endpoints might or might not be included in the interval depending on whether the interval is open, closed, or half- open. 5.2.5: Solving Quadratic Inequalities2

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The solutions to a quadratic inequality can be one interval or two intervals. Use these solutions to create regions on a number line and test points in each region to solve the inequality. If the quadratic equation has only complex solutions, the expression is either always positive or always negative. In these cases, the inequality will have no solution or infinitely many solutions. 5.2.5: Solving Quadratic Inequalities3

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Solutions of quadratic inequalities are often graphed on number lines. The endpoints of the solution interval are represented by either an open dot or a closed dot. Graph the endpoints as an open dot if the original inequality symbol is. Graph endpoints as a closed dot if the original inequality symbol is or . 5.2.5: Solving Quadratic Inequalities4

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For what values of x is ( x 2)( x + 10) > 0? 5.2.5: Solving Quadratic Inequalities5

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The expression will be positive when both factors are positive or both factors are negative. 5.2.5: Solving Quadratic Inequalities6

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x 2 is positive when x > 2. x + 10 is positive when x > 10. Both factors are positive when x > 2 and x > 10, or when x > 2. 5.2.5: Solving Quadratic Inequalities7

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x 2 is negative when x < 2. x + 10 is negative when x < 10. Both factors are negative when x < 2 and x < 10, or when x < 10. ( x 2)( x + 10) > 0 when x > 2 or x < 10. 5.2.5: Solving Quadratic Inequalities8

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Solve x 2 + 8 x + 7 0. Graph the solutions on a number line. 5.2.5: Solving Quadratic Inequalities9

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Ms. Dambreville