Solve the Inequality for x (|x+6|)/8<=1

$\frac{| x + 6 |}{8} \leq 1$

Multiply both sides by $8$ .

$\frac{| x + 6 |}{8} \cdot 8 \leq 1 \cdot 8$

Simplify.

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{| x + 6 |}{8} \cdot 8 \leq 1 \cdot 8$

Rewrite the expression.

$| x + 6 | \leq 1 \cdot 8$

Simplify the right side.

Multiply $8$ by $1$ .

$| x + 6 | \leq 8$

Solve for $x$ .

Write $| x + 6 | \leq 8$ as a piecewise.

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

$x + 6 \geq 0$

Subtract $6$ from both sides of the inequality.

$x \geq - 6$

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

$x + 6 \leq 8$

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 $- 1$ .

$- (x + 6) \leq 8$

Write as a piecewise.

${\begin{cases} x + 6 \leq 8 & x \geq - 6 \\ - (x + 6) \leq 8 & x < - 6 \end{cases}$

Simplify $- (x + 6) \leq 8$ .

Apply the distributive property.

${\begin{cases} x + 6 \leq 8 & x \geq - 6 \\ - x - 1 \cdot 6 \leq 8 & x < - 6 \end{cases}$

Multiply $- 1$ by $6$ .

${\begin{cases} x + 6 \leq 8 & x \geq - 6 \\ - x - 6 \leq 8 & x < - 6 \end{cases}$

Solve $x + 6 \leq 8$ when $x \geq - 6$ .

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

Subtract $6$ from both sides of the inequality.

$x \leq 8 - 6$

Subtract $6$ from $8$ .

$x \leq 2$

Find the intersection of $x \leq 2$ and $x \geq - 6$ .

$- 6 \leq x \leq 2$

Solve $- x - 6 \leq 8$ when $x < - 6$ .

Solve $- x - 6 \leq 8$ for $x$ .

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

Add $6$ to both sides of the inequality.

$- x \leq 8 + 6$

Add $8$ and $6$ .

$- x \leq 14$

Divide each term in $- x \leq 14$ by $- 1$ and simplify.

Divide each term in $- x \leq 14$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{14}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{14}{- 1}$

Divide $x$ by $1$ .

$x \geq \frac{14}{- 1}$

Simplify the right side.

Divide $14$ by $- 1$ .

$x \geq - 14$

Find the intersection of $x \geq - 14$ and $x < - 6$ .

$- 14 \leq x < - 6$

Find the union of the solutions.

$- 14 \leq x \leq 2$

The result can be shown in multiple forms.

Inequality Form:

$- 14 \leq x \leq 2$

Interval Notation:

$[- 14, 2]$

Solve the Inequality for x (|x+6|)/8<=1

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