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Solving Quadratic Inequalities

Solving Quadratic Inequalities. Adapted from Walch Education. Quadratic Inequalities. 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.

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Solving Quadratic Inequalities

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  1. Solving Quadratic Inequalities Adapted from Walch Education

  2. Quadratic Inequalities • Quadratic inequalities can be written in the form ax2+ bx + c < 0, ax2+ bx + c ≤ 0, ax2+ bx + c > 0, or ax2+ 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 Inequalities

  3. Key Concepts • 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 Inequalities

  4. Key Concepts, continued • 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 < or >. • Graph endpoints as a closed dot if the original inequality symbol is ≤ or ≥. 5.2.5: Solving Quadratic Inequalities

  5. Practice # 1 • For what values of x is (x – 2)(x + 10) > 0? 5.2.5: Solving Quadratic Inequalities

  6. Determine the sign possibilities • The expression will be positive when both factors are positive or both factors are negative. 5.2.5: Solving Quadratic Inequalities

  7. Determine when both factors are positive • 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 Inequalities

  8. Determine when both factors are negative • 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 Inequalities

  9. Your Turn • Solve x2+ 8x + 7 ≤ 0. Graph the solutions on a number line. 5.2.5: Solving Quadratic Inequalities

  10. Thanks for Watching! Ms. Dambreville

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