# College Algebra : Rational Inequalities

## Example Questions

### Example Question #1 : Rational Inequalities

Solve

Explanation:

In order to solve this, we need to figure out what values of x makes the numerator and denominator equal zero.

These values are , and .

Then test a few numbers greater or lesser than the above values to figure out what range of values make the inequality true.

### Example Question #1 : Rational Inequalities

Give the solution set of the equation

Explanation:

The boundary points of a rational inequality are the zeroes of the numerator and the denominator.

First, set the numerator equal to 0 and solve for :

Now set the denominator equal to 0 and solve for :

Factor the trinomial using the reverse-FOIL method. Look for two integers whose sum is  and whose product is 7; these are , and , so the equation can be rewritten as

Set each factor to zero and solve for :

Therefore, the boundary points are , which divide the real numbers into four intervals. Choose any value from each interval as a test point, setting  to that value and determining whether the inequality is true.

The four intervals are listed below, along with their arbitrary test points.

: Set

True; include .

: Set

False; exclude .

: Set :

True; include

: Set :

False; exclude .

Since the inequality symbol is the "is greater than or equal to"  symbol, include the zero, 0, of the numerator - but not the other two boundaries, the zeroes of the denominator. This makes the solution set

.