Solve the Absolute Value Inequality for x |x|+7>16

$| x | + 7 > 16$

Write $| x | + 7 > 16$ as a piecewise.

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

$x \geq 0$

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

$x + 7 > 16$

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

$x < 0$

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

$- x + 7 > 16$

Write as a piecewise.

${\begin{cases} x + 7 > 16 & x \geq 0 \\ - x + 7 > 16 & x < 0 \end{cases}$

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

Subtract $7$ from both sides of the inequality.

$x > 16 - 7$

Subtract $7$ from $16$ .

$x > 9$

Solve $- x + 7 > 16$ for $x$ .

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

Subtract $7$ from both sides of the inequality.

$- x > 16 - 7$

Subtract $7$ from $16$ .

$- x > 9$

Divide each term in $- x > 9$ by $- 1$ and simplify.

Divide each term in $- x > 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{9}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{9}{- 1}$

Divide $x$ by $1$ .

$x < \frac{9}{- 1}$

Simplify the right side.

Divide $9$ by $- 1$ .

$x < - 9$

Find the union of the solutions.

$x < - 9$ or $x > 9$

The result can be shown in multiple forms.

Inequality Form:

$x < - 9 or x > 9$

Interval Notation:

$(- \infty, - 9) \cup (9, \infty)$

Solve the Absolute Value Inequality for x |x|+7>16

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