Solve for x 4|3-3x|-6<30

$4 | 3 - 3 x | - 6 < 30$

Write $4 | 3 - 3 x | - 6 < 30$ 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.

$4 (3 - 3 x) - 6 < 30$

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 $- 1$ .

$4 (- (3 - 3 x)) - 6 < 30$

Write as a piecewise.

${\begin{cases} 4 (3 - 3 x) - 6 < 30 & x \leq 1 \\ 4 (- (3 - 3 x)) - 6 < 30 & x > 1 \end{cases}$

Simplify $4 (3 - 3 x) - 6 < 30$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 4 \cdot 3 + 4 (- 3 x) - 6 < 30 & x \leq 1 \\ 4 (- (3 - 3 x)) - 6 < 30 & x > 1 \end{cases}$

Multiply $4$ by $3$ .

${\begin{cases} 12 + 4 (- 3 x) - 6 < 30 & x \leq 1 \\ 4 (- (3 - 3 x)) - 6 < 30 & x > 1 \end{cases}$

Multiply $- 3$ by $4$ .

${\begin{cases} 12 - 12 x - 6 < 30 & x \leq 1 \\ 4 (- (3 - 3 x)) - 6 < 30 & x > 1 \end{cases}$

Subtract $6$ from $12$ .

${\begin{cases} - 12 x + 6 < 30 & x \leq 1 \\ 4 (- (3 - 3 x)) - 6 < 30 & x > 1 \end{cases}$

Simplify $4 (- (3 - 3 x)) - 6 < 30$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 12 x + 6 < 30 & x \leq 1 \\ 4 (- 1 \cdot 3 - (- 3 x)) - 6 < 30 & x > 1 \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} - 12 x + 6 < 30 & x \leq 1 \\ 4 (- 3 - (- 3 x)) - 6 < 30 & x > 1 \end{cases}$

Multiply $- 3$ by $- 1$ .

${\begin{cases} - 12 x + 6 < 30 & x \leq 1 \\ 4 (- 3 + 3 x) - 6 < 30 & x > 1 \end{cases}$

Apply the distributive property.

${\begin{cases} - 12 x + 6 < 30 & x \leq 1 \\ 4 \cdot - 3 + 4 (3 x) - 6 < 30 & x > 1 \end{cases}$

Multiply $4$ by $- 3$ .

${\begin{cases} - 12 x + 6 < 30 & x \leq 1 \\ - 12 + 4 (3 x) - 6 < 30 & x > 1 \end{cases}$

Multiply $3$ by $4$ .

${\begin{cases} - 12 x + 6 < 30 & x \leq 1 \\ - 12 + 12 x - 6 < 30 & x > 1 \end{cases}$

Subtract $6$ from $- 12$ .

${\begin{cases} - 12 x + 6 < 30 & x \leq 1 \\ 12 x - 18 < 30 & x > 1 \end{cases}$

Solve $- 12 x + 6 < 30$ when $x \leq 1$ .

Solve $- 12 x + 6 < 30$ for $x$ .

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

Subtract $6$ from both sides of the inequality.

$- 12 x < 30 - 6$

Subtract $6$ from $30$ .

$- 12 x < 24$

Divide each term in $- 12 x < 24$ by $- 12$ and simplify.

Divide each term in $- 12 x < 24$ by $- 12$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 12 x}{- 12} > \frac{24}{- 12}$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 x}{- 12} > \frac{24}{- 12}$

Divide $x$ by $1$ .

$x > \frac{24}{- 12}$

Simplify the right side.

Divide $24$ by $- 12$ .

$x > - 2$

Find the intersection of $x > - 2$ and $x \leq 1$ .

$- 2 < x \leq 1$

Solve $12 x - 18 < 30$ when $x > 1$ .

Solve $12 x - 18 < 30$ for $x$ .

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

Add $18$ to both sides of the inequality.

$12 x < 30 + 18$

Add $30$ and $18$ .

$12 x < 48$

Divide each term in $12 x < 48$ by $12$ and simplify.

Divide each term in $12 x < 48$ by $12$ .

$\frac{12 x}{12} < \frac{48}{12}$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 x}{12} < \frac{48}{12}$

Divide $x$ by $1$ .

$x < \frac{48}{12}$

Simplify the right side.

Divide $48$ by $12$ .

$x < 4$

Find the intersection of $x < 4$ and $x > 1$ .

$1 < x < 4$

Find the union of the solutions.

$- 2 < x < 4$

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < 4$

Interval Notation:

$(- 2, 4)$

Solve for x 4|3-3x|-6<30

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