Inequalities with Fractions

Inequalities with fractions are solved in a similar manner as quadratic inequalities.

EXAMPLE

Solve .

Solution

First solve the corresponding equation.

The solution, x = 3, determines the intervals on the number line where the fraction may change from greater than 1 to less than 1. This change also may occur on either side of a number that makes the denominator equal 0. Here, the x-value that makes the denominator 0 is x = 0. Test each of the three intervals determined by the numbers 0 and 3.

The symbol means “is not greater than or equal to”. Testing the endpoints 0 and 3 shows that the solution is .

CAUTION

A common error is to try to solve the inequality in the previous example by multiplying both sides by x. The reason this is wrong is that we don’t know in the beginning whether x is positive or negative. If x is negative, the would change to according to one of the properties of inequalities.

EXAMPLE

Solve

Solution

Solve the corresponding equation.

Setting the denominator equal to 0 gives x = 0, so the intervals of interest are . Testing a number from each region in the original inequality and checking the endpoints, we find the solution is

CAUTION

Remember to solve the equation formed by setting the denominator equal to zero. Any number that makes the denominator zero always creates two intervals on the number line. For instance, in the previous example, 0 makes the denominator of the rational inequality equal to 0, so we know that there may be a sign change from one side of 0 to the other (as was indeed the case).

EXAMPLE

Solve .

Solution

Solve the corresponding equation.

Now set the denominator equal to 0 and solve that equation.

x - 9 = 0

(x + 3)(x - 3) = 0

x = 3 or x = -3

or The intervals determined by the three (different) solutions are , Testing a number from each interval in the given inequality shows that the solution is

For this example, none of the endpoints are part of the solution because x = 3 and x = -3 force the denominator to be zero and x = -4 produces an equality.