Solve the Inequality for x |2-3x|>=7

$| 2 - 3 x | \geq 7$

Write $| 2 - 3 x | \geq 7$ as a piecewise.

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

$2 - 3 x \geq 0$

Solve the inequality.

Subtract $2$ from both sides of the inequality.

$- 3 x \geq - 2$

Divide each term in $- 3 x \geq - 2$ by $- 3$ and simplify.

Divide each term in $- 3 x \geq - 2$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} \leq \frac{- 2}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} \leq \frac{- 2}{- 3}$

Divide $x$ by $1$ .

$x \leq \frac{- 2}{- 3}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x \leq \frac{2}{3}$

In the piece where $2 - 3 x$ is non-negative, remove the absolute value.

$2 - 3 x \geq 7$

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

$2 - 3 x < 0$

Solve the inequality.

Subtract $2$ from both sides of the inequality.

$- 3 x < - 2$

Divide each term in $- 3 x < - 2$ by $- 3$ and simplify.

Divide each term in $- 3 x < - 2$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} > \frac{- 2}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} > \frac{- 2}{- 3}$

Divide $x$ by $1$ .

$x > \frac{- 2}{- 3}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x > \frac{2}{3}$

In the piece where $2 - 3 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 - 3 x) \geq 7$

Write as a piecewise.

${\begin{cases} 2 - 3 x \geq 7 & x \leq \frac{2}{3} \\ - (2 - 3 x) \geq 7 & x > \frac{2}{3} \end{cases}$

Simplify $- (2 - 3 x) \geq 7$ .

Apply the distributive property.

${\begin{cases} 2 - 3 x \geq 7 & x \leq \frac{2}{3} \\ - 1 \cdot 2 - (- 3 x) \geq 7 & x > \frac{2}{3} \end{cases}$

Multiply $- 1$ by $2$ .

${\begin{cases} 2 - 3 x \geq 7 & x \leq \frac{2}{3} \\ - 2 - (- 3 x) \geq 7 & x > \frac{2}{3} \end{cases}$

Multiply $- 3$ by $- 1$ .

${\begin{cases} 2 - 3 x \geq 7 & x \leq \frac{2}{3} \\ - 2 + 3 x \geq 7 & x > \frac{2}{3} \end{cases}$

Solve $2 - 3 x \geq 7$ for $x$ .

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

Subtract $2$ from both sides of the inequality.

$- 3 x \geq 7 - 2$

Subtract $2$ from $7$ .

$- 3 x \geq 5$

Divide each term in $- 3 x \geq 5$ by $- 3$ and simplify.

Divide each term in $- 3 x \geq 5$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} \leq \frac{5}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} \leq \frac{5}{- 3}$

Divide $x$ by $1$ .

$x \leq \frac{5}{- 3}$

Simplify the right side.

Move the negative in front of the fraction.

$x \leq - \frac{5}{3}$

Solve $- 2 + 3 x \geq 7$ for $x$ .

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

Add $2$ to both sides of the inequality.

$3 x \geq 7 + 2$

Add $7$ and $2$ .

$3 x \geq 9$

Divide each term in $3 x \geq 9$ by $3$ and simplify.

Divide each term in $3 x \geq 9$ by $3$ .

$\frac{3 x}{3} \geq \frac{9}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{9}{3}$

Divide $x$ by $1$ .

$x \geq \frac{9}{3}$

Simplify the right side.

Divide $9$ by $3$ .

$x \geq 3$

Find the union of the solutions.

$x \leq - \frac{5}{3}$ or $x \geq 3$

The result can be shown in multiple forms.

Inequality Form:

$x \leq - \frac{5}{3} or x \geq 3$

Interval Notation:

$(- \infty, - \frac{5}{3}] \cup [3, \infty)$

Solve the Inequality for x |2-3x|>=7

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