### All Precalculus Resources

## Example Questions

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

What is the solution to the following inequality?

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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:

**Possible Answers:**

**Correct answer:**

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:

**Possible Answers:**

**Correct answer:**

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 .