Solve the Absolute Value Inequality for x |x+4|<=8

$| x + 4 | \leq 8$

Step 1

Write $| x + 4 | \leq 8$ as a piecewise.

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

$x + 4 \geq 0$

Subtract $4$ from both sides of the inequality.

$x \geq - 4$

In the piece where $x + 4$ is non-negative, remove the absolute value.

$x + 4 \leq 8$

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

$x + 4 < 0$

Subtract $4$ from both sides of the inequality.

$x < - 4$

In the piece where $x + 4$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x + 4) \leq 8$

Write as a piecewise.

${\begin{cases} x + 4 \leq 8 & x \geq - 4 \\ - (x + 4) \leq 8 & x < - 4 \end{cases}$

Simplify $- (x + 4) \leq 8$ .

Apply the distributive property.

${\begin{cases} x + 4 \leq 8 & x \geq - 4 \\ - x - 1 \cdot 4 \leq 8 & x < - 4 \end{cases}$

Multiply $- 1$ by $4$ .

${\begin{cases} x + 4 \leq 8 & x \geq - 4 \\ - x - 4 \leq 8 & x < - 4 \end{cases}$

Step 2

Solve $x + 4 \leq 8$ when $x \geq - 4$ .

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

Subtract $4$ from both sides of the inequality.

$x \leq 8 - 4$

Subtract $4$ from $8$ .

$x \leq 4$

Find the intersection of $x \leq 4$ and $x \geq - 4$ .

$- 4 \leq x \leq 4$

Step 3

Solve $- x - 4 \leq 8$ when $x < - 4$ .

Solve $- x - 4 \leq 8$ for $x$ .

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

Add $4$ to both sides of the inequality.

$- x \leq 8 + 4$

Add $8$ and $4$ .

$- x \leq 12$

Divide each term in $- x \leq 12$ by $- 1$ and simplify.

Divide each term in $- x \leq 12$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{12}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{12}{- 1}$

Divide $x$ by $1$ .

$x \geq \frac{12}{- 1}$

Simplify the right side.

Divide $12$ by $- 1$ .

$x \geq - 12$

Find the intersection of $x \geq - 12$ and $x < - 4$ .

$- 12 \leq x < - 4$

Step 4

Find the union of the solutions.

$- 12 \leq x \leq 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 12 \leq x \leq 4$

Interval Notation:

$[- 12, 4]$

Step 6

Solve the Absolute Value Inequality for x |x+4|<=8

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