How do you solve inequalities with fractions?

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.

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.