To find the interval for the first piece, find where the inside of the absolute value is non-negative.

Solve the inequality.

Combine and .

Add to both sides of the inequality.

Multiply both sides of the equation by .

Multiply by .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

In the piece where is non-negative, remove the absolute value.

To find the interval for the second piece, find where the inside of the absolute value is negative.

Solve the inequality.

Combine and .

Add to both sides of the inequality.

Multiply both sides of the equation by .

Multiply by .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

In the piece where is negative, remove the absolute value and multiply by .

Write as a piecewise.

Combine and .

Simplify .

Combine and .

Apply the distributive property.

Multiply by .

Solve for .

Move all terms not containing to the right side of the inequality.

Add to both sides of the inequality.

Add and .

Multiply both sides of the equation by .

Multiply by .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

Find the intersection of and .

Solve for .

Move all terms not containing to the right side of the inequality.

Subtract from both sides of the inequality.

Subtract from .

Multiply each term in by

Multiply each term in by . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

Multiply .

Multiply by .

Multiply by .

Multiply by .

Multiply both sides of the equation by .

Multiply by .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

Find the intersection of and .

Find the union of the solutions.

or

The result can be shown in multiple forms.

Inequality Form:

Interval Notation:

Solve for x |4/5x-4|>8