# Precalculus : Solve and Graph Rational Inequalities

## Example Questions

### Example Question #1 : Solve And Graph Rational Inequalities

Solve the inequality.       Explanation:

First, subtract from both sides so you get .

Then find the common denominator and simplify .

Next, factor out the numerator and set each of the three factor equal to zero and solve for .

The solutions are .

Now plug in values between   , and into the inequality and observe if the conditions of the inequality are met.

Note that . They are met in the interval and .

Thus, the solution to the inequality is ### Example Question #1 : Solve And Graph Rational Inequalities

Solve and graph:       Explanation:

1) Multiply both sides of the equation by the common denominator of the fractions:  2) Simplify:     3) For standard notation, and the fact that inequalities can be read backwards: For interval notation: 4) Graph: ### Example Question #1 : Solving Polynomial And Rational Inequalities

Solve and graph:       Explanation:

Graph the rational expression, 1) Because and a divide by is undefined in the real number system, there is a vertical asymptote where .

2) As   ,   , and as  ,  .

3) As  ,  , and as   ,  .

4) The funtion y is exists over the allowed x-intervals: One approach for solving the inequality:

For 1) Determine where over the x-values or .

2) for the intervals or .

3) Then the solution is .

Another approach for solving the inequality:

1) Write as , then determine the x-values that cause to be true: 2) is true for or .

3) Then the solution is .

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