# Precalculus Examples

Step 1

Subtract from both sides of the inequality.

Step 2

Step 2.1

Factor out of .

Step 2.1.1

Factor out of .

Step 2.1.2

Factor out of .

Step 2.1.3

Factor out of .

Step 2.2

To write as a fraction with a common denominator, multiply by .

Step 2.3

Combine and .

Step 2.4

Combine the numerators over the common denominator.

Step 2.5

Simplify the numerator.

Step 2.5.1

Apply the distributive property.

Step 2.5.2

Multiply by .

Step 2.5.3

Subtract from .

Step 3

Find all the values where the expression switches from negative to positive by setting each factor equal to and solving.

Step 4

Subtract from both sides of the equation.

Step 5

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

Step 6

Consolidate the solutions.

Step 7

Step 7.1

Set the denominator in equal to to find where the expression is undefined.

Step 7.2

The domain is all values of that make the expression defined.

Step 8

Use each root to create test intervals.

Step 9

Step 9.1

Test a value on the interval to see if it makes the inequality true.

Step 9.1.1

Choose a value on the interval and see if this value makes the original inequality true.

Step 9.1.2

Replace with in the original inequality.

Step 9.1.3

The left side is not less than the right side , which means that the given statement is false.

False

False

Step 9.2

Test a value on the interval to see if it makes the inequality true.

Step 9.2.1

Choose a value on the interval and see if this value makes the original inequality true.

Step 9.2.2

Replace with in the original inequality.

Step 9.2.3

The left side is less than the right side , which means that the given statement is always true.

True

True

Step 9.3

Test a value on the interval to see if it makes the inequality true.

Step 9.3.1

Choose a value on the interval and see if this value makes the original inequality true.

Step 9.3.2

Replace with in the original inequality.

Step 9.3.3

The left side is not less than the right side , which means that the given statement is false.

False

False

Step 9.4

Compare the intervals to determine which ones satisfy the original inequality.

False

True

False

False

True

False

Step 10

The solution consists of all of the true intervals.

Step 11

The result can be shown in multiple forms.

Inequality Form:

Interval Notation:

Step 12