Solve for x -|6+3x|-8<-11

$- | 6 + 3 x | - 8 < - 11$

Write $- | 6 + 3 x | - 8 < - 11$ as a piecewise.

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

$6 + 3 x \geq 0$

Solve the inequality.

Subtract $6$ from both sides of the inequality.

$3 x \geq - 6$

Divide each term in $3 x \geq - 6$ by $3$ and simplify.

Divide each term in $3 x \geq - 6$ by $3$ .

$\frac{3 x}{3} \geq \frac{- 6}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{- 6}{3}$

Divide $x$ by $1$ .

$x \geq \frac{- 6}{3}$

Simplify the right side.

Divide $- 6$ by $3$ .

$x \geq - 2$

In the piece where $6 + 3 x$ is non-negative, remove the absolute value.

$- (6 + 3 x) - 8 < - 11$

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

$6 + 3 x < 0$

Solve the inequality.

Subtract $6$ from both sides of the inequality.

$3 x < - 6$

Divide each term in $3 x < - 6$ by $3$ and simplify.

Divide each term in $3 x < - 6$ by $3$ .

$\frac{3 x}{3} < \frac{- 6}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{- 6}{3}$

Divide $x$ by $1$ .

$x < \frac{- 6}{3}$

Simplify the right side.

Divide $- 6$ by $3$ .

$x < - 2$

In the piece where $6 + 3 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- - (6 + 3 x) - 8 < - 11$

Write as a piecewise.

${\begin{cases} - (6 + 3 x) - 8 < - 11 & x \geq - 2 \\ - - (6 + 3 x) - 8 < - 11 & x < - 2 \end{cases}$

Simplify $- (6 + 3 x) - 8 < - 11$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 1 \cdot 6 - (3 x) - 8 < - 11 & x \geq - 2 \\ - - (6 + 3 x) - 8 < - 11 & x < - 2 \end{cases}$

Multiply $- 1$ by $6$ .

${\begin{cases} - 6 - (3 x) - 8 < - 11 & x \geq - 2 \\ - - (6 + 3 x) - 8 < - 11 & x < - 2 \end{cases}$

Multiply $3$ by $- 1$ .

${\begin{cases} - 6 - 3 x - 8 < - 11 & x \geq - 2 \\ - - (6 + 3 x) - 8 < - 11 & x < - 2 \end{cases}$

Subtract $8$ from $- 6$ .

${\begin{cases} - 3 x - 14 < - 11 & x \geq - 2 \\ - - (6 + 3 x) - 8 < - 11 & x < - 2 \end{cases}$

Simplify $- - (6 + 3 x) - 8 < - 11$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 3 x - 14 < - 11 & x \geq - 2 \\ - (- 1 \cdot 6 - (3 x)) - 8 < - 11 & x < - 2 \end{cases}$

Multiply $- 1$ by $6$ .

${\begin{cases} - 3 x - 14 < - 11 & x \geq - 2 \\ - (- 6 - (3 x)) - 8 < - 11 & x < - 2 \end{cases}$

Multiply $3$ by $- 1$ .

${\begin{cases} - 3 x - 14 < - 11 & x \geq - 2 \\ - (- 6 - 3 x) - 8 < - 11 & x < - 2 \end{cases}$

Apply the distributive property.

${\begin{cases} - 3 x - 14 < - 11 & x \geq - 2 \\ - - 6 - (- 3 x) - 8 < - 11 & x < - 2 \end{cases}$

Multiply $- 1$ by $- 6$ .

${\begin{cases} - 3 x - 14 < - 11 & x \geq - 2 \\ 6 - (- 3 x) - 8 < - 11 & x < - 2 \end{cases}$

Multiply $- 3$ by $- 1$ .

${\begin{cases} - 3 x - 14 < - 11 & x \geq - 2 \\ 6 + 3 x - 8 < - 11 & x < - 2 \end{cases}$

Subtract $8$ from $6$ .

${\begin{cases} - 3 x - 14 < - 11 & x \geq - 2 \\ 3 x - 2 < - 11 & x < - 2 \end{cases}$

Solve $- 3 x - 14 < - 11$ for $x$ .

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

Add $14$ to both sides of the inequality.

$- 3 x < - 11 + 14$

Add $- 11$ and $14$ .

$- 3 x < 3$

Divide each term in $- 3 x < 3$ by $- 3$ and simplify.

Divide each term in $- 3 x < 3$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} > \frac{3}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} > \frac{3}{- 3}$

Divide $x$ by $1$ .

$x > \frac{3}{- 3}$

Simplify the right side.

Divide $3$ by $- 3$ .

$x > - 1$

Solve $3 x - 2 < - 11$ for $x$ .

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

Add $2$ to both sides of the inequality.

$3 x < - 11 + 2$

Add $- 11$ and $2$ .

$3 x < - 9$

Divide each term in $3 x < - 9$ by $3$ and simplify.

Divide each term in $3 x < - 9$ by $3$ .

$\frac{3 x}{3} < \frac{- 9}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{- 9}{3}$

Divide $x$ by $1$ .

$x < \frac{- 9}{3}$

Simplify the right side.

Divide $- 9$ by $3$ .

$x < - 3$

Find the union of the solutions.

$x < - 3$ or $x > - 1$

The result can be shown in multiple forms.

Inequality Form:

$x < - 3 or x > - 1$

Interval Notation:

$(- \infty, - 3) \cup (- 1, \infty)$

Solve for x -|6+3x|-8<-11

Related Questions

Support LCMGCF.com by disabling your adblocker.