Solve for x -2|3-3x|-2>=-14

$- 2 | 3 - 3 x | - 2 \geq - 14$

Write $- 2 | 3 - 3 x | - 2 \geq - 14$ as a piecewise.

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

$3 - 3 x \geq 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$- 3 x \geq - 3$

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

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

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

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 3$ by $- 3$ .

$x \leq 1$

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

$- 2 (3 - 3 x) - 2 \geq - 14$

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

$3 - 3 x < 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$- 3 x < - 3$

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 $- 3$ .

$x > 1$

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

$- 2 (- (3 - 3 x)) - 2 \geq - 14$

Write as a piecewise.

${\begin{cases} - 2 (3 - 3 x) - 2 \geq - 14 & x \leq 1 \\ - 2 (- (3 - 3 x)) - 2 \geq - 14 & x > 1 \end{cases}$

Simplify $- 2 (3 - 3 x) - 2 \geq - 14$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 2 \cdot 3 - 2 (- 3 x) - 2 \geq - 14 & x \leq 1 \\ - 2 (- (3 - 3 x)) - 2 \geq - 14 & x > 1 \end{cases}$

Multiply $- 2$ by $3$ .

${\begin{cases} - 6 - 2 (- 3 x) - 2 \geq - 14 & x \leq 1 \\ - 2 (- (3 - 3 x)) - 2 \geq - 14 & x > 1 \end{cases}$

Multiply $- 3$ by $- 2$ .

${\begin{cases} - 6 + 6 x - 2 \geq - 14 & x \leq 1 \\ - 2 (- (3 - 3 x)) - 2 \geq - 14 & x > 1 \end{cases}$

Subtract $2$ from $- 6$ .

${\begin{cases} 6 x - 8 \geq - 14 & x \leq 1 \\ - 2 (- (3 - 3 x)) - 2 \geq - 14 & x > 1 \end{cases}$

Simplify $- 2 (- (3 - 3 x)) - 2 \geq - 14$ .

Simplify each term.

Apply the distributive property.

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

Multiply $- 1$ by $3$ .

${\begin{cases} 6 x - 8 \geq - 14 & x \leq 1 \\ - 2 (- 3 - (- 3 x)) - 2 \geq - 14 & x > 1 \end{cases}$

Multiply $- 3$ by $- 1$ .

${\begin{cases} 6 x - 8 \geq - 14 & x \leq 1 \\ - 2 (- 3 + 3 x) - 2 \geq - 14 & x > 1 \end{cases}$

Apply the distributive property.

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

Multiply $- 2$ by $- 3$ .

${\begin{cases} 6 x - 8 \geq - 14 & x \leq 1 \\ 6 - 2 (3 x) - 2 \geq - 14 & x > 1 \end{cases}$

Multiply $3$ by $- 2$ .

${\begin{cases} 6 x - 8 \geq - 14 & x \leq 1 \\ 6 - 6 x - 2 \geq - 14 & x > 1 \end{cases}$

Subtract $2$ from $6$ .

${\begin{cases} 6 x - 8 \geq - 14 & x \leq 1 \\ - 6 x + 4 \geq - 14 & x > 1 \end{cases}$

Solve $6 x - 8 \geq - 14$ when $x \leq 1$ .

Solve $6 x - 8 \geq - 14$ for $x$ .

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

Add $8$ to both sides of the inequality.

$6 x \geq - 14 + 8$

Add $- 14$ and $8$ .

$6 x \geq - 6$

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

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

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

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 6$ by $6$ .

$x \geq - 1$

Find the intersection of $x \geq - 1$ and $x \leq 1$ .

$- 1 \leq x \leq 1$

Solve $- 6 x + 4 \geq - 14$ when $x > 1$ .

Solve $- 6 x + 4 \geq - 14$ for $x$ .

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

Subtract $4$ from both sides of the inequality.

$- 6 x \geq - 14 - 4$

Subtract $4$ from $- 14$ .

$- 6 x \geq - 18$

Divide each term in $- 6 x \geq - 18$ by $- 6$ and simplify.

Divide each term in $- 6 x \geq - 18$ by $- 6$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 6 x}{- 6} \leq \frac{- 18}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 x}{- 6} \leq \frac{- 18}{- 6}$

Divide $x$ by $1$ .

$x \leq \frac{- 18}{- 6}$

Simplify the right side.

Divide $- 18$ by $- 6$ .

$x \leq 3$

Find the intersection of $x \leq 3$ and $x > 1$ .

$1 < x \leq 3$

Find the union of the solutions.

$- 1 \leq x \leq 3$

The result can be shown in multiple forms.

Inequality Form:

$- 1 \leq x \leq 3$

Interval Notation:

$[- 1, 3]$

Solve for x -2|3-3x|-2>=-14

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