Solve for x -2|x+3|<-8

$- 2 | x + 3 | < - 8$

Write $- 2 | x + 3 | < - 8$ as a piecewise.

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

$x + 3 \geq 0$

Subtract $3$ from both sides of the inequality.

$x \geq - 3$

In the piece where $x + 3$ is non-negative, remove the absolute value.

$- 2 (x + 3) < - 8$

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

$x + 3 < 0$

Subtract $3$ from both sides of the inequality.

$x < - 3$

In the piece where $x + 3$ is negative, remove the absolute value and multiply by $- 1$ .

$- 2 (- (x + 3)) < - 8$

Write as a piecewise.

${\begin{cases} - 2 (x + 3) < - 8 & x \geq - 3 \\ - 2 (- (x + 3)) < - 8 & x < - 3 \end{cases}$

Simplify $- 2 (x + 3) < - 8$ .

Apply the distributive property.

${\begin{cases} - 2 x - 2 \cdot 3 < - 8 & x \geq - 3 \\ - 2 (- (x + 3)) < - 8 & x < - 3 \end{cases}$

Multiply $- 2$ by $3$ .

${\begin{cases} - 2 x - 6 < - 8 & x \geq - 3 \\ - 2 (- (x + 3)) < - 8 & x < - 3 \end{cases}$

Simplify $- 2 (- (x + 3)) < - 8$ .

Apply the distributive property.

${\begin{cases} - 2 x - 6 < - 8 & x \geq - 3 \\ - 2 (- x - 1 \cdot 3) < - 8 & x < - 3 \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} - 2 x - 6 < - 8 & x \geq - 3 \\ - 2 (- x - 3) < - 8 & x < - 3 \end{cases}$

Apply the distributive property.

${\begin{cases} - 2 x - 6 < - 8 & x \geq - 3 \\ - 2 (- x) - 2 \cdot - 3 < - 8 & x < - 3 \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} - 2 x - 6 < - 8 & x \geq - 3 \\ 2 x - 2 \cdot - 3 < - 8 & x < - 3 \end{cases}$

Multiply $- 2$ by $- 3$ .

${\begin{cases} - 2 x - 6 < - 8 & x \geq - 3 \\ 2 x + 6 < - 8 & x < - 3 \end{cases}$

Solve $- 2 x - 6 < - 8$ for $x$ .

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

Add $6$ to both sides of the inequality.

$- 2 x < - 8 + 6$

Add $- 8$ and $6$ .

$- 2 x < - 2$

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

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

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 2$ by $- 2$ .

$x > 1$

Solve $2 x + 6 < - 8$ for $x$ .

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

Subtract $6$ from both sides of the inequality.

$2 x < - 8 - 6$

Subtract $6$ from $- 8$ .

$2 x < - 14$

Divide each term in $2 x < - 14$ by $2$ and simplify.

Divide each term in $2 x < - 14$ by $2$ .

$\frac{2 x}{2} < \frac{- 14}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{- 14}{2}$

Divide $x$ by $1$ .

$x < \frac{- 14}{2}$

Simplify the right side.

Divide $- 14$ by $2$ .

$x < - 7$

Find the union of the solutions.

$x < - 7$ or $x > 1$

The result can be shown in multiple forms.

Inequality Form:

$x < - 7 or x > 1$

Interval Notation:

$(- \infty, - 7) \cup (1, \infty)$

Solve for x -2|x+3|<-8

Related Questions

Support LCMGCF.com by disabling your adblocker.