Solve the Inequality for x |-x-10|+4<-8

$| - x - 10 | + 4 < - 8$

Step 1

Write $| - x - 10 | + 4 < - 8$ as a piecewise.

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

$- x - 10 \geq 0$

Solve the inequality.

Add $10$ to both sides of the inequality.

$- x \geq 10$

Divide each term in $- x \geq 10$ by $- 1$ and simplify.

Divide each term in $- x \geq 10$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{10}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{10}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{10}{- 1}$

Simplify the right side.

Divide $10$ by $- 1$ .

$x \leq - 10$

In the piece where $- x - 10$ is non-negative, remove the absolute value.

$- x - 10 + 4 < - 8$

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

$- x - 10 < 0$

Solve the inequality.

Add $10$ to both sides of the inequality.

$- x < 10$

Divide each term in $- x < 10$ by $- 1$ and simplify.

Divide each term in $- x < 10$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{10}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{10}{- 1}$

Divide $x$ by $1$ .

$x > \frac{10}{- 1}$

Simplify the right side.

Divide $10$ by $- 1$ .

$x > - 10$

In the piece where $- x - 10$ is negative, remove the absolute value and multiply by $- 1$ .

$- (- x - 10) + 4 < - 8$

Write as a piecewise.

${\begin{cases} - x - 10 + 4 < - 8 & x \leq - 10 \\ - (- x - 10) + 4 < - 8 & x > - 10 \end{cases}$

Add $- 10$ and $4$ .

${\begin{cases} - x - 6 < - 8 & x \leq - 10 \\ - (- x - 10) + 4 < - 8 & x > - 10 \end{cases}$

Simplify $- (- x - 10) + 4 < - 8$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - x - 6 < - 8 & x \leq - 10 \\ - - x - - 10 + 4 < - 8 & x > - 10 \end{cases}$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

${\begin{cases} - x - 6 < - 8 & x \leq - 10 \\ 1 x - - 10 + 4 < - 8 & x > - 10 \end{cases}$

Multiply $x$ by $1$ .

${\begin{cases} - x - 6 < - 8 & x \leq - 10 \\ x - - 10 + 4 < - 8 & x > - 10 \end{cases}$

Multiply $- 1$ by $- 10$ .

${\begin{cases} - x - 6 < - 8 & x \leq - 10 \\ x + 10 + 4 < - 8 & x > - 10 \end{cases}$

Add $10$ and $4$ .

${\begin{cases} - x - 6 < - 8 & x \leq - 10 \\ x + 14 < - 8 & x > - 10 \end{cases}$

Step 2

Solve $- x - 6 < - 8$ when $x \leq - 10$ .

Solve $- x - 6 < - 8$ for $x$ .

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

Add $6$ to both sides of the inequality.

$- x < - 8 + 6$

Add $- 8$ and $6$ .

$- x < - 2$

Divide each term in $- x < - 2$ by $- 1$ and simplify.

Divide each term in $- x < - 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{- 2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{- 2}{- 1}$

Divide $x$ by $1$ .

$x > \frac{- 2}{- 1}$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$x > 2$

Find the intersection of $x > 2$ and $x \leq - 10$ .

No solution

Step 3

Solve $x + 14 < - 8$ when $x > - 10$ .

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

Subtract $14$ from both sides of the inequality.

$x < - 8 - 14$

Subtract $14$ from $- 8$ .

$x < - 22$

Find the intersection of $x < - 22$ and $x > - 10$ .

No solution

Step 4

Find the union of the solutions.

No solution

Solve the Inequality for x |-x-10|+4<-8

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