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How do you solve inequalities with fractions?

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Quick Answer

Solving for a variable in inequalities is very similar to solving for a variable in an equation, except that you are looking for a range of values instead of just one. To solve inequalities with fractions, or rational inequalities, find the critical values and test values in between them.

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Full Answer

  1. Find the critical values

    In inequalities involving fractions, it's important to take note when the inequality switches from true to false or there is a zero-denominator. Solve for the inequality as if it were an equation and solve for zero-denominators. For example, the starting inequality might be (x + 3)/(x - 2) > -2 Rewrite it as (x + 3)/(x - 2) = -2 and solve for x. (x + 3)/(x - 2) = -2 (x - 2)*(x + 3)/(x - 2) = -2*(x - 2) x + 3 = -2x + 4 x + 2x = 4 - 3 3x = 1 x = 1/3 Solve for x - 2 = 0. At x = 2, there is a zero denominator. The critical points are 1/3 and 2.

  2. Test values near critical values

    Plug values into the original inequality. If the critical points are 1/3 and 2, test 0, 1, 3. x = 0, (0 + 3)/(0 - 2) > -2 is true because -3/2 is greater than -2; x = 1, (1 + 3)/(1 - 2) > -2 is false because -4 is less than -2; x = 3, (3 + 3)/(3 - 2) > -2 is true because 6 is greater than -2.

  3. Express the variable's range of values

    The variable's possible values are written as an inequality. For (x + 3)/(x - 2) > -2, x can greater than 2 or less than 1/3. x < 1/3, x > 2.

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