# Calculus Examples

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

Since does not contain the variable to solve for, move it to the right side of the equation by adding to both sides.

Since does not contain the variable to solve for, move it to the right side of the equation by subtracting from both sides.

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

Use each root to create test intervals.

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

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

Replace with in the original inequality.

Simplify.

Simplify the numerator.

Remove parentheses around .

Subtract from to get .

Simplify the denominator.

Remove parentheses.

Remove parentheses around .

Add and to get .

Divide by to get .

The left side is greater than the right side , which means the given statement is true.

True

True

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

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

Replace with in the original inequality.

Simplify.

Simplify the numerator.

Remove parentheses around .

Subtract from to get .

Simplify the denominator.

Remove parentheses.

Remove parentheses around .

Add and to get .

Divide by to get .

The left side is less than the right side , which means the given statement is false.

False

False

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

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

Replace with in the original inequality.

Simplify.

Simplify the numerator.

Remove parentheses around .

Subtract from to get .

Simplify the denominator.

Remove parentheses.

Remove parentheses around .

Add and to get .

The left side is greater than the right side , which means the given statement is true.

True

True

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

True

False

True

True

False

True

The solution consists of all of the true intervals.

Remove any values from the solution that make the denominator equal to .

or