Solve the Inequality for x (|x+9|)/4>2

$\frac{| x + 9 |}{4} > 2$

Multiply both sides by $4$ .

$\frac{| x + 9 |}{4} \cdot 4 > 2 \cdot 4$

Simplify.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{| x + 9 |}{4} \cdot 4 > 2 \cdot 4$

Rewrite the expression.

$| x + 9 | > 2 \cdot 4$

Simplify the right side.

Multiply $2$ by $4$ .

$| x + 9 | > 8$

Solve for $x$ .

Write $| x + 9 | > 8$ as a piecewise.

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

$x + 9 \geq 0$

Subtract $9$ from both sides of the inequality.

$x \geq - 9$

In the piece where $x + 9$ is non-negative, remove the absolute value.

$x + 9 > 8$

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

$x + 9 < 0$

Subtract $9$ from both sides of the inequality.

$x < - 9$

In the piece where $x + 9$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x + 9) > 8$

Write as a piecewise.

${\begin{cases} x + 9 > 8 & x \geq - 9 \\ - (x + 9) > 8 & x < - 9 \end{cases}$

Simplify $- (x + 9) > 8$ .

Apply the distributive property.

${\begin{cases} x + 9 > 8 & x \geq - 9 \\ - x - 1 \cdot 9 > 8 & x < - 9 \end{cases}$

Multiply $- 1$ by $9$ .

${\begin{cases} x + 9 > 8 & x \geq - 9 \\ - x - 9 > 8 & x < - 9 \end{cases}$

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

Subtract $9$ from both sides of the inequality.

$x > 8 - 9$

Subtract $9$ from $8$ .

$x > - 1$

Solve $- x - 9 > 8$ for $x$ .

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

Add $9$ to both sides of the inequality.

$- x > 8 + 9$

Add $8$ and $9$ .

$- x > 17$

Divide each term in $- x > 17$ by $- 1$ and simplify.

Divide each term in $- x > 17$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{17}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{17}{- 1}$

Divide $x$ by $1$ .

$x < \frac{17}{- 1}$

Simplify the right side.

Divide $17$ by $- 1$ .

$x < - 17$

Find the union of the solutions.

$x < - 17$ or $x > - 1$

The result can be shown in multiple forms.

Inequality Form:

$x < - 17 or x > - 1$

Interval Notation:

$(- \infty, - 17) \cup (- 1, \infty)$

Solve the Inequality for x (|x+9|)/4>2

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