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1.7

1.7. Inequalities Part 2. Quadratic Inequalities Rational Inequalities. SOLVING A THREE-PART INEQUALITY. Bellringer. Solve – 2 < 5 + 3 x < 20. Solution. Subtract 5 from each part. Divide each part by 3. Quadratic Inequalities.

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1.7

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  1. 1.7 Inequalities Part 2 Quadratic Inequalities Rational Inequalities

  2. SOLVING A THREE-PART INEQUALITY Bellringer Solve –2 < 5 + 3x < 20. Solution Subtract 5 from each part. Divide each part by 3.

  3. Quadratic Inequalities A quadratic inequality is an inequality that can be written in the form for real numbers a, b, and c with a ≠ 0. (The symbol < can be replaced with >, ≤, or ≥.)

  4. Solving a Quadratic Inequality Step 1 Solve the corresponding quadratic equation. Step 2 Identify the intervals determined by the solutions of the equation. Step 3 Use a test value from each interval to determine which intervals form the solution set.

  5. SOLVING A QUADRATIC INEQULITY Example 5 Solve

  6. SOLVING QUADRATIC INEQULAITY Example 6 Solve

  7. Solving a Rational Inequality Step 1 Rewrite the inequality, if necessary, so that 0 is on one side and there is a single fraction on the other side. Step 2 Determine the values that will cause either the numerator or the denominator of the rational expression to equal 0. These values determine the intervals of the number line to consider.

  8. Solving a Rational Inequality Step 3 Use a test value from each interval to determine which intervals form the solution set. A value causing the denominator to equal zero will never be included in the solution set. If the inequality is strict, any value causing the numerator to equal zero will be excluded; if nonstrict, any such value will be included.

  9. Caution Solving a rational inequality such as by multiplying both sides by x + 4 to obtain 5 ≥ x + 4 requires considering two cases, since the sign of x + 4 depends on the value of x. If x + 4 were negative, then the inequality symbol must be reversed. The procedure described in the next two examples eliminates the need for considering separate cases.

  10. SOLVING A RATIONAL INEQUALITY Example 8 Solve

  11. SOLVING A RATIONAL INEQUALITY Example 8 Solve Solution

  12. SOLVING A RATIONAL INEQUALITY Example 8 Solve

  13. Caution As suggested by Example 8, be careful with the endpoints of the intervals when solving rational inequalities.

  14. SOLVING A RATIONAL INEQULAITY Example 9 Solve

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