Solve the Inequality for x -7|x+12|<=-14

$- 7 | x + 12 | \leq - 14$

Step 1

Write $- 7 | x + 12 | \leq - 14$ as a piecewise.

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

$x + 12 \geq 0$

Subtract $12$ from both sides of the inequality.

$x \geq - 12$

In the piece where $x + 12$ is non-negative, remove the absolute value.

$- 7 (x + 12) \leq - 14$

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

$x + 12 < 0$

Subtract $12$ from both sides of the inequality.

$x < - 12$

In the piece where $x + 12$ is negative, remove the absolute value and multiply by $- 1$ .

$- 7 (- (x + 12)) \leq - 14$

Write as a piecewise.

${\begin{cases} - 7 (x + 12) \leq - 14 & x \geq - 12 \\ - 7 (- (x + 12)) \leq - 14 & x < - 12 \end{cases}$

Simplify $- 7 (x + 12) \leq - 14$ .

Apply the distributive property.

${\begin{cases} - 7 x - 7 \cdot 12 \leq - 14 & x \geq - 12 \\ - 7 (- (x + 12)) \leq - 14 & x < - 12 \end{cases}$

Multiply $- 7$ by $12$ .

${\begin{cases} - 7 x - 84 \leq - 14 & x \geq - 12 \\ - 7 (- (x + 12)) \leq - 14 & x < - 12 \end{cases}$

Simplify $- 7 (- (x + 12)) \leq - 14$ .

Apply the distributive property.

${\begin{cases} - 7 x - 84 \leq - 14 & x \geq - 12 \\ - 7 (- x - 1 \cdot 12) \leq - 14 & x < - 12 \end{cases}$

Multiply $- 1$ by $12$ .

${\begin{cases} - 7 x - 84 \leq - 14 & x \geq - 12 \\ - 7 (- x - 12) \leq - 14 & x < - 12 \end{cases}$

Apply the distributive property.

${\begin{cases} - 7 x - 84 \leq - 14 & x \geq - 12 \\ - 7 (- x) - 7 \cdot - 12 \leq - 14 & x < - 12 \end{cases}$

Multiply $- 1$ by $- 7$ .

${\begin{cases} - 7 x - 84 \leq - 14 & x \geq - 12 \\ 7 x - 7 \cdot - 12 \leq - 14 & x < - 12 \end{cases}$

Multiply $- 7$ by $- 12$ .

${\begin{cases} - 7 x - 84 \leq - 14 & x \geq - 12 \\ 7 x + 84 \leq - 14 & x < - 12 \end{cases}$

Step 2

Solve $- 7 x - 84 \leq - 14$ for $x$ .

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

Add $84$ to both sides of the inequality.

$- 7 x \leq - 14 + 84$

Add $- 14$ and $84$ .

$- 7 x \leq 70$

Divide each term in $- 7 x \leq 70$ by $- 7$ and simplify.

Divide each term in $- 7 x \leq 70$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} \geq \frac{70}{- 7}$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 x}{- 7} \geq \frac{70}{- 7}$

Divide $x$ by $1$ .

$x \geq \frac{70}{- 7}$

Simplify the right side.

Divide $70$ by $- 7$ .

$x \geq - 10$

Step 3

Solve $7 x + 84 \leq - 14$ for $x$ .

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

Subtract $84$ from both sides of the inequality.

$7 x \leq - 14 - 84$

Subtract $84$ from $- 14$ .

$7 x \leq - 98$

Divide each term in $7 x \leq - 98$ by $7$ and simplify.

Divide each term in $7 x \leq - 98$ by $7$ .

$\frac{7 x}{7} \leq \frac{- 98}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} \leq \frac{- 98}{7}$

Divide $x$ by $1$ .

$x \leq \frac{- 98}{7}$

Simplify the right side.

Divide $- 98$ by $7$ .

$x \leq - 14$

Step 4

Find the union of the solutions.

$x \leq - 14$ or $x \geq - 10$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 14 or x \geq - 10$

Interval Notation:

$(- \infty, - 14] \cup [- 10, \infty)$

Step 6

Solve the Inequality for x -7|x+12|<=-14

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