Subtract <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> from both sides of the inequality.

To write <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> as a fraction with a common denominator, multiply by <math><mstyle displaystyle="true"><mfrac><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac></mstyle></math> .

Combine the numerators over the common denominator.

Simplify the numerator.

Apply the distributive property.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

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

Factor <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mo>-</mo><mi>x</mi></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mo>-</mo><mn>8</mn></mstyle></math> as <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mo>-</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>1</mn><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mo>-</mo><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>8</mn><mo>)</mo></mrow></mstyle></math> as <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>8</mn><mo>)</mo></mrow></mstyle></math> .

Move the negative in front of the fraction.

Find all the values where the expression switches from negative to positive by setting each factor equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and solving.

Subtract <math><mstyle displaystyle="true"><mn>8</mn></mstyle></math> from both sides of the equation.

Subtract <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> from both sides of the equation.

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

Consolidate the solutions.

Set the denominator in <math><mstyle displaystyle="true"><mfrac><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> to find where the expression is undefined.

Subtract <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> from both sides of the equation.

The domain is all values of <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> that make the expression defined.

Use each root to create test intervals.

Test a value on the interval <math><mstyle displaystyle="true"><mi>x</mi><mo><</mo><mo>-</mo><mn>8</mn></mstyle></math> to see if it makes the inequality true.

Choose a value on the interval <math><mstyle displaystyle="true"><mi>x</mi><mo><</mo><mo>-</mo><mn>8</mn></mstyle></math> and see if this value makes the original inequality true.

Replace <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mn>10</mn></mstyle></math> in the original inequality.

The left side <math><mstyle displaystyle="true"><mn><mn>1.</mn><mover accent="true"><mn>6</mn><mo>‾</mo></mover></mn></mstyle></math> is less than the right side <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> , which means that the given statement is false.

False

False

Test a value on the interval <math><mstyle displaystyle="true"><mo>-</mo><mn>8</mn><mo><</mo><mi>x</mi><mo><</mo><mo>-</mo><mn>4</mn></mstyle></math> to see if it makes the inequality true.

Choose a value on the interval <math><mstyle displaystyle="true"><mo>-</mo><mn>8</mn><mo><</mo><mi>x</mi><mo><</mo><mo>-</mo><mn>4</mn></mstyle></math> and see if this value makes the original inequality true.

Replace <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mn>6</mn></mstyle></math> in the original inequality.

The left side <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> is greater than the right side <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> , which means that the given statement is always true.

True

True

Test a value on the interval <math><mstyle displaystyle="true"><mi>x</mi><mo>></mo><mo>-</mo><mn>4</mn></mstyle></math> to see if it makes the inequality true.

Choose a value on the interval <math><mstyle displaystyle="true"><mi>x</mi><mo>></mo><mo>-</mo><mn>4</mn></mstyle></math> and see if this value makes the original inequality true.

Replace <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> in the original inequality.

The left side <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> is less than the right side <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> , which means that the given statement is false.

False

False

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

The solution consists of all of the true intervals.

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