# Precalculus : Solving Polynomial and Rational Inequalities

## Example Questions

### Example Question #21 : Inequalities And Linear Programming

What is the solution to the following inequality?

Explanation:

First, we must solve for the roots of the cubic polynomial equation.

We obtain that the roots are .

Now there are four regions created by these numbers:

• . In this region, the values of the polynomial are negative (i.e.plug in  and you obtain

• . In this region, the values of the polynomial are positive (when , polynomial evaluates to )

•  . In this region the polynomial switches again to negative.

• . In this region the values of the polynomial are positive

Hence the two regions we want are  and .

### 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 #22 : Inequalities And Linear Programming

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 #2 : Solve And Graph 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 .