Solve the Absolute Value Inequality for x |x+9|<=7

$| x + 9 | \leq 7$

Step 1

Write $| x + 9 | \leq 7$ 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.

$x + 9 \leq 7$

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 $- 1$ .

$- (x + 9) \leq 7$

Write as a piecewise.

${\begin{cases} x + 9 \leq 7 & x \geq - 9 \\ - (x + 9) \leq 7 & x < - 9 \end{cases}$

Simplify $- (x + 9) \leq 7$ .

Apply the distributive property.

${\begin{cases} x + 9 \leq 7 & x \geq - 9 \\ - x - 1 \cdot 9 \leq 7 & x < - 9 \end{cases}$

Multiply $- 1$ by $9$ .

${\begin{cases} x + 9 \leq 7 & x \geq - 9 \\ - x - 9 \leq 7 & x < - 9 \end{cases}$

Step 2

Solve $x + 9 \leq 7$ when $x \geq - 9$ .

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

Subtract $9$ from both sides of the inequality.

$x \leq 7 - 9$

Subtract $9$ from $7$ .

$x \leq - 2$

Find the intersection of $x \leq - 2$ and $x \geq - 9$ .

$- 9 \leq x \leq - 2$

Step 3

Solve $- x - 9 \leq 7$ when $x < - 9$ .

Solve $- x - 9 \leq 7$ for $x$ .

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

Add $9$ to both sides of the inequality.

$- x \leq 7 + 9$

Add $7$ and $9$ .

$- x \leq 16$

Divide each term in $- x \leq 16$ by $- 1$ and simplify.

Divide each term in $- x \leq 16$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{16}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{16}{- 1}$

Divide $x$ by $1$ .

$x \geq \frac{16}{- 1}$

Simplify the right side.

Divide $16$ by $- 1$ .

$x \geq - 16$

Find the intersection of $x \geq - 16$ and $x < - 9$ .

$- 16 \leq x < - 9$

Step 4

Find the union of the solutions.

$- 16 \leq x \leq - 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 16 \leq x \leq - 2$

Interval Notation:

$[- 16, - 2]$

Step 6

Solve the Absolute Value Inequality for x |x+9|<=7

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