Rational Functions: Solving Rational Equations and Inequalities

Solving Rational Inequalities Using Algebra

You can also use algebra to solve rational inequalities.

For example, solve 1 divided by 4 a plus 5 divided by 8 a is greater than one-half .

First rewrite the inequality as 1 divided by 4 a plus 5 divided by 8 a minus one-half is greater than 0 .

Next solve for the zeros.

Next find the excluded values. In this case the excluded value is a = 0. This value will make the denominator equal to zero.

Next identify these points on a number line.

Number line with a equal zero and a equal one and three-fourths marked

Next check values in the three intervals, a < 0, 0 < a < one and three-fourths , a > . Pick any value that fits within the interval to see if the inequality holds true. If one value in the interval is true, then all values in the interval will work, and vice-versa.

Try a = -1.

This statement is not true, therefore no values in the interval a < 0 are true.

Next try a = 1.

This statement is true. Therefore every value in the interval 0 < a < one and three-fourths will be true.

Finally, check a = 2.

This statement is not true, therefore no values in the interval a > one and three-fourths are true.

The final answer is 0 < a < one and three-fourths .