Solve for x |3x-6|+3<15

$| 3 x - 6 | + 3 < 15$

Step 1

Write $| 3 x - 6 | + 3 < 15$ as a piecewise.

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

$3 x - 6 \geq 0$

Solve the inequality.

Add $6$ to 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 $3 x - 6$ is non-negative, remove the absolute value.

$3 x - 6 + 3 < 15$

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

$3 x - 6 < 0$

Solve the inequality.

Add $6$ to 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 $3 x - 6$ is negative, remove the absolute value and multiply by $- 1$ .

$- (3 x - 6) + 3 < 15$

Write as a piecewise.

${\begin{cases} 3 x - 6 + 3 < 15 & x \geq 2 \\ - (3 x - 6) + 3 < 15 & x < 2 \end{cases}$

Add $- 6$ and $3$ .

${\begin{cases} 3 x - 3 < 15 & x \geq 2 \\ - (3 x - 6) + 3 < 15 & x < 2 \end{cases}$

Simplify $- (3 x - 6) + 3 < 15$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 3 x - 3 < 15 & x \geq 2 \\ - (3 x) - - 6 + 3 < 15 & x < 2 \end{cases}$

Multiply $3$ by $- 1$ .

${\begin{cases} 3 x - 3 < 15 & x \geq 2 \\ - 3 x - - 6 + 3 < 15 & x < 2 \end{cases}$

Multiply $- 1$ by $- 6$ .

${\begin{cases} 3 x - 3 < 15 & x \geq 2 \\ - 3 x + 6 + 3 < 15 & x < 2 \end{cases}$

Add $6$ and $3$ .

${\begin{cases} 3 x - 3 < 15 & x \geq 2 \\ - 3 x + 9 < 15 & x < 2 \end{cases}$

Step 2

Solve $3 x - 3 < 15$ when $x \geq 2$ .

Solve $3 x - 3 < 15$ for $x$ .

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

Add $3$ to both sides of the inequality.

$3 x < 15 + 3$

Add $15$ and $3$ .

$3 x < 18$

Divide each term in $3 x < 18$ by $3$ and simplify.

Divide each term in $3 x < 18$ by $3$ .

$\frac{3 x}{3} < \frac{18}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{18}{3}$

Divide $x$ by $1$ .

$x < \frac{18}{3}$

Simplify the right side.

Divide $18$ by $3$ .

$x < 6$

Find the intersection of $x < 6$ and $x \geq 2$ .

$2 \leq x < 6$

Step 3

Solve $- 3 x + 9 < 15$ when $x < 2$ .

Solve $- 3 x + 9 < 15$ for $x$ .

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

Subtract $9$ from both sides of the inequality.

$- 3 x < 15 - 9$

Subtract $9$ from $15$ .

$- 3 x < 6$

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 $x < 2$ .

$- 2 < x < 2$

Step 4

Find the union of the solutions.

$- 2 < x < 6$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < 6$

Interval Notation:

$(- 2, 6)$

Step 6

Solve for x |3x-6|+3<15

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