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## Example Questions

### Example Question #1 : Rational Inequalities

Solve

**Possible Answers:**

**Correct answer:**

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 #2 : Rational Inequalities

Give the solution set of the equation

**Possible Answers:**

**Correct answer:**

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

.

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