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

Solve the inequality.

Add to both sides of the inequality.

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.

Add to both sides of the inequality.

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.

Add and .

Simplify .

Simplify each term.

Apply the distributive property.

Multiply by .

Multiply by .

Add and .

Solve for .

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

Add to both sides of the inequality.

Add and .

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 .

Divide each term by and simplify.

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

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

Find the intersection of and .

Find the union of the solutions.

The result can be shown in multiple forms.

Inequality Form:

Interval Notation:

Solve for x |7x-7|+3<24