Precalculus Examples Absolute Value Expressions and Equations

Write the Absolute Value as Piecewise

$| 4 x - 3 |$

Step 1

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

$4 x - 3 \geq 0$

Step 2

Solve the inequality.

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Step 2.1

Add $3$ to both sides of the inequality.

$4 x \geq 3$

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3

In the piece where $4 x - 3$ is non-negative, remove the absolute value.

$4 x - 3$

Step 4

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

$4 x - 3 < 0$

Step 5

Solve the inequality.

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Step 5.1

Add $3$ to both sides of the inequality.

$4 x < 3$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $4$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $x$ by $1$ .

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

Step 6

In the piece where $4 x - 3$ is negative, remove the absolute value and multiply by $- 1$ .

$- (4 x - 3)$

Step 7

Write as a piecewise.

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

Step 8

Simplify $- (4 x - 3)$ .

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Step 8.1

Apply the distributive property.

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

Step 8.2

Multiply $4$ by $- 1$ .

${\begin{cases} 4 x - 3 & x \geq \frac{3}{4} \\ - 4 x - - 3 & x < \frac{3}{4} \end{cases}$

Step 8.3

Multiply $- 1$ by $- 3$ .

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

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Write the Absolute Value as Piecewise

Absolute Value Expressions and Equations

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