Solve the Absolute Value Inequality for x |x+6|>7

$| x + 6 | > 7$

Step 1

Write $| x + 6 | > 7$ as a piecewise.

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

$x + 6 \geq 0$

Subtract $6$ from both sides of the inequality.

$x \geq - 6$

In the piece where $x + 6$ is non-negative, remove the absolute value.

$x + 6 > 7$

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

$x + 6 < 0$

Subtract $6$ from both sides of the inequality.

$x < - 6$

In the piece where $x + 6$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x + 6) > 7$

Write as a piecewise.

${\begin{cases} x + 6 > 7 & x \geq - 6 \\ - (x + 6) > 7 & x < - 6 \end{cases}$

Simplify $- (x + 6) > 7$ .

Apply the distributive property.

${\begin{cases} x + 6 > 7 & x \geq - 6 \\ - x - 1 \cdot 6 > 7 & x < - 6 \end{cases}$

Multiply $- 1$ by $6$ .

${\begin{cases} x + 6 > 7 & x \geq - 6 \\ - x - 6 > 7 & x < - 6 \end{cases}$

Step 2

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

Subtract $6$ from both sides of the inequality.

$x > 7 - 6$

Subtract $6$ from $7$ .

$x > 1$

Step 3

Solve $- x - 6 > 7$ for $x$ .

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

Add $6$ to both sides of the inequality.

$- x > 7 + 6$

Add $7$ and $6$ .

$- x > 13$

Divide each term in $- x > 13$ by $- 1$ and simplify.

Divide each term in $- x > 13$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{13}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{13}{- 1}$

Divide $x$ by $1$ .

$x < \frac{13}{- 1}$

Simplify the right side.

Divide $13$ by $- 1$ .

$x < - 13$

Step 4

Find the union of the solutions.

$x < - 13$ or $x > 1$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < - 13 or x > 1$

Interval Notation:

$(- \infty, - 13) \cup (1, \infty)$

Step 6

Solve the Absolute Value Inequality for x |x+6|>7

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