Solve the Inequality for x |3/2x-2|+1<=2

$| \frac{3}{2} x - 2 | + 1 \leq 2$

Step 1

Write $| \frac{3}{2} x - 2 | + 1 \leq 2$ as a piecewise.

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

$\frac{3}{2} x - 2 \geq 0$

Solve the inequality.

Combine $\frac{3}{2}$ and $x$ .

$\frac{3 x}{2} - 2 \geq 0$

Add $2$ to both sides of the inequality.

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

Multiply both sides by $2$ .

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

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$3 x \geq 2 \cdot 2$

Simplify the right side.

Multiply $2$ by $2$ .

$3 x \geq 4$

Divide each term in $3 x \geq 4$ by $3$ and simplify.

Divide each term in $3 x \geq 4$ by $3$ .

$\frac{3 x}{3} \geq \frac{4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{4}{3}$

Divide $x$ by $1$ .

$x \geq \frac{4}{3}$

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

$\frac{3}{2} x - 2 + 1 \leq 2$

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

$\frac{3}{2} x - 2 < 0$

Solve the inequality.

Combine $\frac{3}{2}$ and $x$ .

$\frac{3 x}{2} - 2 < 0$

Add $2$ to both sides of the inequality.

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

Multiply both sides by $2$ .

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

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$3 x < 2 \cdot 2$

Simplify the right side.

Multiply $2$ by $2$ .

$3 x < 4$

Divide each term in $3 x < 4$ by $3$ and simplify.

Divide each term in $3 x < 4$ by $3$ .

$\frac{3 x}{3} < \frac{4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{4}{3}$

Divide $x$ by $1$ .

$x < \frac{4}{3}$

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

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

Write as a piecewise.

${\begin{cases} \frac{3}{2} x - 2 + 1 \leq 2 & x \geq \frac{4}{3} \\ - (\frac{3}{2} x - 2) + 1 \leq 2 & x < \frac{4}{3} \end{cases}$

Simplify $\frac{3}{2} x - 2 + 1 \leq 2$ .

Combine $\frac{3}{2}$ and $x$ .

${\begin{cases} \frac{3 x}{2} - 2 + 1 \leq 2 & x \geq \frac{4}{3} \\ - (\frac{3}{2} x - 2) + 1 \leq 2 & x < \frac{4}{3} \end{cases}$

Add $- 2$ and $1$ .

${\begin{cases} \frac{3 x}{2} - 1 \leq 2 & x \geq \frac{4}{3} \\ - (\frac{3}{2} x - 2) + 1 \leq 2 & x < \frac{4}{3} \end{cases}$

Simplify $- (\frac{3}{2} x - 2) + 1 \leq 2$ .

Simplify each term.

Combine $\frac{3}{2}$ and $x$ .

${\begin{cases} \frac{3 x}{2} - 1 \leq 2 & x \geq \frac{4}{3} \\ - (\frac{3 x}{2} - 2) + 1 \leq 2 & x < \frac{4}{3} \end{cases}$

Apply the distributive property.

${\begin{cases} \frac{3 x}{2} - 1 \leq 2 & x \geq \frac{4}{3} \\ - \frac{3 x}{2} - - 2 + 1 \leq 2 & x < \frac{4}{3} \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} \frac{3 x}{2} - 1 \leq 2 & x \geq \frac{4}{3} \\ - \frac{3 x}{2} + 2 + 1 \leq 2 & x < \frac{4}{3} \end{cases}$

Add $2$ and $1$ .

${\begin{cases} \frac{3 x}{2} - 1 \leq 2 & x \geq \frac{4}{3} \\ - \frac{3 x}{2} + 3 \leq 2 & x < \frac{4}{3} \end{cases}$

Step 2

Solve $\frac{3 x}{2} - 1 \leq 2$ when $x \geq \frac{4}{3}$ .

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

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

Add $1$ to both sides of the inequality.

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

Add $2$ and $1$ .

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

Multiply both sides by $2$ .

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

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$3 x \leq 3 \cdot 2$

Simplify the right side.

Multiply $3$ by $2$ .

$3 x \leq 6$

Divide each term in $3 x \leq 6$ by $3$ and simplify.

Divide each term in $3 x \leq 6$ by $3$ .

$\frac{3 x}{3} \leq \frac{6}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \leq \frac{6}{3}$

Divide $x$ by $1$ .

$x \leq \frac{6}{3}$

Simplify the right side.

Divide $6$ by $3$ .

$x \leq 2$

Find the intersection of $x \leq 2$ and $x \geq \frac{4}{3}$ .

$\frac{4}{3} \leq x \leq 2$

Step 3

Solve $- \frac{3 x}{2} + 3 \leq 2$ when $x < \frac{4}{3}$ .

Solve $- \frac{3 x}{2} + 3 \leq 2$ for $x$ .

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

Subtract $3$ from both sides of the inequality.

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

Subtract $3$ from $2$ .

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

Divide each term in $- \frac{3 x}{2} \leq - 1$ by $- 1$ and simplify.

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

$\frac{- \frac{3 x}{2}}{- 1} \geq \frac{- 1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{3 x}{2}}{1} \geq \frac{- 1}{- 1}$

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

$\frac{3 x}{2} \geq \frac{- 1}{- 1}$

Simplify the right side.

Divide $- 1$ by $- 1$ .

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

Multiply both sides by $2$ .

$\frac{3 x}{2} \cdot 2 \geq 1 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 x}{2} \cdot 2 \geq 1 \cdot 2$

Rewrite the expression.

$3 x \geq 1 \cdot 2$

Simplify the right side.

Multiply $2$ by $1$ .

$3 x \geq 2$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Find the intersection of $x \geq \frac{2}{3}$ and $x < \frac{4}{3}$ .

$\frac{2}{3} \leq x < \frac{4}{3}$

Step 4

Find the union of the solutions.

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

Step 5

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$[\frac{2}{3}, 2]$

Step 6

Solve the Inequality for x |3/2x-2|+1<=2

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