Precalculus Examples

$f (x) = | 2 x - 3 |$

Step 1

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

$2 x - 3 \geq 0$

Step 2

Solve the inequality.

Tap for more steps...

Step 2.1

Add $3$ to both sides of the inequality.

$2 x \geq 3$

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3

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

$2 x - 3$

Step 4

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

$2 x - 3 < 0$

Step 5

Solve the inequality.

Tap for more steps...

Step 5.1

Add $3$ to both sides of the inequality.

$2 x < 3$

Step 5.2

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

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $x$ by $1$ .

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

Step 6

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

$- (2 x - 3)$

Step 7

Write as a piecewise.

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

Step 8

Simplify $- (2 x - 3)$ .

Tap for more steps...

Step 8.1

Apply the distributive property.

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

Step 8.2

Multiply $2$ by $- 1$ .

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

Step 8.3

Multiply $- 1$ by $- 3$ .

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