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

Solve the inequality.

Subtract <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> from both sides of the inequality.

Divide each term in <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo>≥</mo><mo>-</mo><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo>≥</mo><mo>-</mo><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Simplify the left side.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Simplify the right side.

Divide <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

In the piece where <math><mstyle displaystyle="true"><mn>3</mn><mo>+</mo><mn>3</mn><mi>x</mi></mstyle></math> 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.

Subtract <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> from both sides of the inequality.

Divide each term in <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo><</mo><mo>-</mo><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo><</mo><mo>-</mo><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Simplify the left side.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Simplify the right side.

Divide <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

In the piece where <math><mstyle displaystyle="true"><mn>3</mn><mo>+</mo><mn>3</mn><mi>x</mi></mstyle></math> is negative, remove the absolute value and multiply by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Write as a piecewise.

Simplify <math><mstyle displaystyle="true"><mo>-</mo><mrow><mo>(</mo><mn>3</mn><mo>+</mo><mn>3</mn><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>2</mn><mo>></mo><mo>-</mo><mn>11</mn></mstyle></math> .

Simplify each term.

Apply the distributive property.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> from <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn></mstyle></math> .

Simplify <math><mstyle displaystyle="true"><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>3</mn><mo>+</mo><mn>3</mn><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>2</mn><mo>></mo><mo>-</mo><mn>11</mn></mstyle></math> .

Simplify each term.

Apply the distributive property.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Apply the distributive property.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Solve <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn><mo>></mo><mo>-</mo><mn>11</mn></mstyle></math> for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Move all terms not containing <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> to the right side of the inequality.

Add <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> to both sides of the inequality.

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>11</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> .

Divide each term in <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn><mi>x</mi><mo>></mo><mo>-</mo><mn>6</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn><mi>x</mi><mo>></mo><mo>-</mo><mn>6</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn></mstyle></math> . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

Simplify the left side.

Cancel the common factor of <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn></mstyle></math> .

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Simplify the right side.

Divide <math><mstyle displaystyle="true"><mo>-</mo><mn>6</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn></mstyle></math> .

Find the intersection of <math><mstyle displaystyle="true"><mi>x</mi><mo><</mo><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mi>x</mi><mo>≥</mo><mo>-</mo><mn>1</mn></mstyle></math> .

Solve <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>></mo><mo>-</mo><mn>11</mn></mstyle></math> for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Move all terms not containing <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> to the right side of the inequality.

Subtract <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> from both sides of the inequality.

Subtract <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> from <math><mstyle displaystyle="true"><mo>-</mo><mn>11</mn></mstyle></math> .

Divide each term in <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo>></mo><mo>-</mo><mn>12</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo>></mo><mo>-</mo><mn>12</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Simplify the left side.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Cancel the common factor.

Simplify the right side.

Divide <math><mstyle displaystyle="true"><mo>-</mo><mn>12</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Find the intersection of <math><mstyle displaystyle="true"><mi>x</mi><mo>></mo><mo>-</mo><mn>4</mn></mstyle></math> and <math><mstyle displaystyle="true"><mi>x</mi><mo><</mo><mo>-</mo><mn>1</mn></mstyle></math> .

Find the union of the solutions.

The result can be shown in multiple forms.

Inequality Form:

Interval Notation:

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