Solve the Inequality for x 0<|2-3x|<1

$0 < | 2 - 3 x | < 1$

Find the $x$ values that make $0 < | 2 - 3 x |$ true.

Rewrite so $x$ is on the left side of the inequality.

$| 2 - 3 x | > 0$

Write $| 2 - 3 x | > 0$ 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.

$2 - 3 x > 0$

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 $- 1$ .

$- (2 - 3 x) > 0$

Write as a piecewise.

${\begin{cases} 2 - 3 x > 0 & x \leq \frac{2}{3} \\ - (2 - 3 x) > 0 & x > \frac{2}{3} \end{cases}$

Simplify $- (2 - 3 x) > 0$ .

Apply the distributive property.

${\begin{cases} 2 - 3 x > 0 & x \leq \frac{2}{3} \\ - 1 \cdot 2 - (- 3 x) > 0 & x > \frac{2}{3} \end{cases}$

Multiply $- 1$ by $2$ .

${\begin{cases} 2 - 3 x > 0 & x \leq \frac{2}{3} \\ - 2 - (- 3 x) > 0 & x > \frac{2}{3} \end{cases}$

Multiply $- 3$ by $- 1$ .

${\begin{cases} 2 - 3 x > 0 & x \leq \frac{2}{3} \\ - 2 + 3 x > 0 & x > \frac{2}{3} \end{cases}$

Solve $2 - 3 x > 0$ for $x$ .

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.

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

Solve $- 2 + 3 x > 0$ for $x$ .

Add $2$ to 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 $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Find the union of the solutions.

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

Find the $x$ values that make $| 2 - 3 x | < 1$ true.

Write $| 2 - 3 x | < 1$ 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.

$2 - 3 x < 1$

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 $- 1$ .

$- (2 - 3 x) < 1$

Write as a piecewise.

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

Simplify $- (2 - 3 x) < 1$ .

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

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

Multiply $- 3$ by $- 1$ .

${\begin{cases} 2 - 3 x < 1 & x \leq \frac{2}{3} \\ - 2 + 3 x < 1 & x > \frac{2}{3} \end{cases}$

Solve $2 - 3 x < 1$ when $x \leq \frac{2}{3}$ .

Solve $2 - 3 x < 1$ for $x$ .

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

Subtract $2$ from both sides of the inequality.

$- 3 x < 1 - 2$

Subtract $2$ from $1$ .

$- 3 x < - 1$

Divide each term in $- 3 x < - 1$ by $- 3$ and simplify.

Divide each term in $- 3 x < - 1$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} > \frac{- 1}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} > \frac{- 1}{- 3}$

Divide $x$ by $1$ .

$x > \frac{- 1}{- 3}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x > \frac{1}{3}$

Find the intersection of $x > \frac{1}{3}$ and $x \leq \frac{2}{3}$ .

$\frac{1}{3} < x \leq \frac{2}{3}$

Solve $- 2 + 3 x < 1$ when $x > \frac{2}{3}$ .

Solve $- 2 + 3 x < 1$ for $x$ .

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

Add $2$ to both sides of the inequality.

$3 x < 1 + 2$

Add $1$ and $2$ .

$3 x < 3$

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 $3$ .

$x < 1$

Find the intersection of $x < 1$ and $x > \frac{2}{3}$ .

$\frac{2}{3} < x < 1$

Find the union of the solutions.

$\frac{1}{3} < x < 1$

The solution is the intersection of the intervals.

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

$\frac{1}{3} < x < 1$

Find the intersection.

$\frac{1}{3} < x < \frac{2}{3}$ or $\frac{2}{3} < x < 1$

The result can be shown in multiple forms.

Inequality Form:

$\frac{1}{3} < x < \frac{2}{3} or \frac{2}{3} < x < 1$

Interval Notation:

$(\frac{1}{3}, \frac{2}{3}) \cup (\frac{2}{3}, 1)$

Solve the Inequality for x 0<|2-3x|<1

Related Questions

Support LCMGCF.com by disabling your adblocker.