Solve for x |(2x+6)/3|<2

$| \frac{2 x + 6}{3} | < 2$

Step 1

Write $| \frac{2 x + 6}{3} | < 2$ as a piecewise.

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

$\frac{2 x + 6}{3} \geq 0$

Solve the inequality.

Multiply both sides by $3$ .

$\frac{2 x + 6}{3} \cdot 3 \geq 0 \cdot 3$

Simplify.

Simplify the left side.

Simplify $\frac{2 x + 6}{3} \cdot 3$ .

Factor $2$ out of $2 x + 6$ .

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 6}{3} \cdot 3 \geq 0 \cdot 3$

Factor $2$ out of $6$ .

$\frac{2 x + 2 \cdot 3}{3} \cdot 3 \geq 0 \cdot 3$

Factor $2$ out of $2 x + 2 \cdot 3$ .

$\frac{2 (x + 3)}{3} \cdot 3 \geq 0 \cdot 3$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{2 (x + 3)}{3} \cdot 3 \geq 0 \cdot 3$

Rewrite the expression.

$2 (x + 3) \geq 0 \cdot 3$

Apply the distributive property.

$2 x + 2 \cdot 3 \geq 0 \cdot 3$

Multiply $2$ by $3$ .

$2 x + 6 \geq 0 \cdot 3$

Simplify the right side.

Multiply $0$ by $3$ .

$2 x + 6 \geq 0$

Solve for $x$ .

Subtract $6$ from both sides of the inequality.

$2 x \geq - 6$

Divide each term in $2 x \geq - 6$ by $2$ and simplify.

Divide each term in $2 x \geq - 6$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 6}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 6}{2}$

Divide $x$ by $1$ .

$x \geq \frac{- 6}{2}$

Simplify the right side.

Divide $- 6$ by $2$ .

$x \geq - 3$

In the piece where $\frac{2 x + 6}{3}$ is non-negative, remove the absolute value.

$\frac{2 x + 6}{3} < 2$

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

$\frac{2 x + 6}{3} < 0$

Solve the inequality.

Multiply both sides by $3$ .

$\frac{2 x + 6}{3} \cdot 3 < 0 \cdot 3$

Simplify.

Simplify the left side.

Simplify $\frac{2 x + 6}{3} \cdot 3$ .

Factor $2$ out of $2 x + 6$ .

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 6}{3} \cdot 3 < 0 \cdot 3$

Factor $2$ out of $6$ .

$\frac{2 x + 2 \cdot 3}{3} \cdot 3 < 0 \cdot 3$

Factor $2$ out of $2 x + 2 \cdot 3$ .

$\frac{2 (x + 3)}{3} \cdot 3 < 0 \cdot 3$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{2 (x + 3)}{3} \cdot 3 < 0 \cdot 3$

Rewrite the expression.

$2 (x + 3) < 0 \cdot 3$

Apply the distributive property.

$2 x + 2 \cdot 3 < 0 \cdot 3$

Multiply $2$ by $3$ .

$2 x + 6 < 0 \cdot 3$

Simplify the right side.

Multiply $0$ by $3$ .

$2 x + 6 < 0$

Solve for $x$ .

Subtract $6$ from both sides of the inequality.

$2 x < - 6$

Divide each term in $2 x < - 6$ by $2$ and simplify.

Divide each term in $2 x < - 6$ by $2$ .

$\frac{2 x}{2} < \frac{- 6}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{- 6}{2}$

Divide $x$ by $1$ .

$x < \frac{- 6}{2}$

Simplify the right side.

Divide $- 6$ by $2$ .

$x < - 3$

In the piece where $\frac{2 x + 6}{3}$ is negative, remove the absolute value and multiply by $- 1$ .

$- \frac{2 x + 6}{3} < 2$

Write as a piecewise.

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

Factor $2$ out of $2 x + 6$ .

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $6$ .

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

Factor $2$ out of $2 x + 2 \cdot 3$ .

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

Factor $2$ out of $2 x + 6$ .

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $6$ .

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

Factor $2$ out of $2 x + 2 \cdot 3$ .

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

Step 2

Solve $\frac{2 (x + 3)}{3} < 2$ when $x \geq - 3$ .

Solve $\frac{2 (x + 3)}{3} < 2$ for $x$ .

Multiply both sides by $3$ .

$\frac{2 (x + 3)}{3} \cdot 3 < 2 \cdot 3$

Simplify.

Simplify the left side.

Simplify $\frac{2 (x + 3)}{3} \cdot 3$ .

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{2 (x + 3)}{3} \cdot 3 < 2 \cdot 3$

Rewrite the expression.

$2 (x + 3) < 2 \cdot 3$

Apply the distributive property.

$2 x + 2 \cdot 3 < 2 \cdot 3$

Multiply $2$ by $3$ .

$2 x + 6 < 2 \cdot 3$

Simplify the right side.

Multiply $2$ by $3$ .

$2 x + 6 < 6$

Solve for $x$ .

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

Subtract $6$ from both sides of the inequality.

$2 x < 6 - 6$

Subtract $6$ from $6$ .

$2 x < 0$

Divide each term in $2 x < 0$ by $2$ and simplify.

Divide each term in $2 x < 0$ by $2$ .

$\frac{2 x}{2} < \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{0}{2}$

Divide $x$ by $1$ .

$x < \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x < 0$

Find the intersection of $x < 0$ and $x \geq - 3$ .

$- 3 \leq x < 0$

Step 3

Solve $- \frac{2 (x + 3)}{3} < 2$ when $x < - 3$ .

Solve $- \frac{2 (x + 3)}{3} < 2$ for $x$ .

Divide each term in $- \frac{2 (x + 3)}{3} < 2$ by $- 1$ and simplify.

Divide each term in $- \frac{2 (x + 3)}{3} < 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{2 (x + 3)}{3}}{- 1} > \frac{2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{2 (x + 3)}{3}}{1} > \frac{2}{- 1}$

Divide $\frac{2 (x + 3)}{3}$ by $1$ .

$\frac{2 (x + 3)}{3} > \frac{2}{- 1}$

Simplify the right side.

Divide $2$ by $- 1$ .

$\frac{2 (x + 3)}{3} > - 2$

Multiply both sides by $3$ .

$\frac{2 (x + 3)}{3} \cdot 3 > - 2 \cdot 3$

Simplify.

Simplify the left side.

Simplify $\frac{2 (x + 3)}{3} \cdot 3$ .

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{2 (x + 3)}{3} \cdot 3 > - 2 \cdot 3$

Rewrite the expression.

$2 (x + 3) > - 2 \cdot 3$

Apply the distributive property.

$2 x + 2 \cdot 3 > - 2 \cdot 3$

Multiply $2$ by $3$ .

$2 x + 6 > - 2 \cdot 3$

Simplify the right side.

Multiply $- 2$ by $3$ .

$2 x + 6 > - 6$

Solve for $x$ .

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

Subtract $6$ from both sides of the inequality.

$2 x > - 6 - 6$

Subtract $6$ from $- 6$ .

$2 x > - 12$

Divide each term in $2 x > - 12$ by $2$ and simplify.

Divide each term in $2 x > - 12$ by $2$ .

$\frac{2 x}{2} > \frac{- 12}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} > \frac{- 12}{2}$

Divide $x$ by $1$ .

$x > \frac{- 12}{2}$

Simplify the right side.

Divide $- 12$ by $2$ .

$x > - 6$

Find the intersection of $x > - 6$ and $x < - 3$ .

$- 6 < x < - 3$

Step 4

Find the union of the solutions.

$- 6 < x < 0$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 6 < x < 0$

Interval Notation:

$(- 6, 0)$

Step 6

Solve for x |(2x+6)/3|<2

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