Solve for x |2x-3|<3

$| 2 x - 3 | < 3$

Step 1

Write $| 2 x - 3 | < 3$ as a piecewise.

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

$2 x - 3 \geq 0$

Solve the inequality.

Add $3$ to both sides of the inequality.

$2 x \geq 3$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

$2 x - 3 < 3$

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

$2 x - 3 < 0$

Solve the inequality.

Add $3$ to both sides of the inequality.

$2 x < 3$

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

Divide each term in $2 x < 3$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

$- (2 x - 3) < 3$

Write as a piecewise.

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

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $- 3$ .

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

Step 2

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

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

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

Add $3$ to both sides of the inequality.

$2 x < 3 + 3$

Add $3$ and $3$ .

$2 x < 6$

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 $2$ .

$x < 3$

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

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

Step 3

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

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

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

Subtract $3$ from both sides of the inequality.

$- 2 x < 3 - 3$

Subtract $3$ from $3$ .

$- 2 x < 0$

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

Divide each term in $- 2 x < 0$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} > \frac{0}{- 2}$

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

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

Step 4

Find the union of the solutions.

$0 < x < 3$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$0 < x < 3$

Interval Notation:

$(0, 3)$

Step 6

Solve for x |2x-3|<3

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