Solve for x |3x-3|+4<13

$| 3 x - 3 | + 4 < 13$

Step 1

Write $| 3 x - 3 | + 4 < 13$ as a piecewise.

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

$3 x - 3 \geq 0$

Solve the inequality.

Add $3$ to 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 $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $3$ by $3$ .

$x \geq 1$

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

$3 x - 3 + 4 < 13$

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

$3 x - 3 < 0$

Solve the inequality.

Add $3$ to 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 $3$ .

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

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

$- (3 x - 3) + 4 < 13$

Write as a piecewise.

${\begin{cases} 3 x - 3 + 4 < 13 & x \geq 1 \\ - (3 x - 3) + 4 < 13 & x < 1 \end{cases}$

Add $- 3$ and $4$ .

${\begin{cases} 3 x + 1 < 13 & x \geq 1 \\ - (3 x - 3) + 4 < 13 & x < 1 \end{cases}$

Simplify $- (3 x - 3) + 4 < 13$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 3 x + 1 < 13 & x \geq 1 \\ - (3 x) - - 3 + 4 < 13 & x < 1 \end{cases}$

Multiply $3$ by $- 1$ .

${\begin{cases} 3 x + 1 < 13 & x \geq 1 \\ - 3 x - - 3 + 4 < 13 & x < 1 \end{cases}$

Multiply $- 1$ by $- 3$ .

${\begin{cases} 3 x + 1 < 13 & x \geq 1 \\ - 3 x + 3 + 4 < 13 & x < 1 \end{cases}$

Add $3$ and $4$ .

${\begin{cases} 3 x + 1 < 13 & x \geq 1 \\ - 3 x + 7 < 13 & x < 1 \end{cases}$

Step 2

Solve $3 x + 1 < 13$ when $x \geq 1$ .

Solve $3 x + 1 < 13$ for $x$ .

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

Subtract $1$ from both sides of the inequality.

$3 x < 13 - 1$

Subtract $1$ from $13$ .

$3 x < 12$

Divide each term in $3 x < 12$ by $3$ and simplify.

Divide each term in $3 x < 12$ by $3$ .

$\frac{3 x}{3} < \frac{12}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{12}{3}$

Divide $x$ by $1$ .

$x < \frac{12}{3}$

Simplify the right side.

Divide $12$ by $3$ .

$x < 4$

Find the intersection of $x < 4$ and $x \geq 1$ .

$1 \leq x < 4$

Step 3

Solve $- 3 x + 7 < 13$ when $x < 1$ .

Solve $- 3 x + 7 < 13$ for $x$ .

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

Subtract $7$ from both sides of the inequality.

$- 3 x < 13 - 7$

Subtract $7$ from $13$ .

$- 3 x < 6$

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

Divide each term in $- 3 x < 6$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

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 $- 3$ .

$x > - 2$

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

$- 2 < x < 1$

Step 4

Find the union of the solutions.

$- 2 < x < 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < 4$

Interval Notation:

$(- 2, 4)$

Step 6

Solve for x |3x-3|+4<13

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