Algebra Examples

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

Step 1

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

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

Step 2

Solve the inequality.

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Step 2.1

Multiply both sides by $2$ .

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

Step 2.2

Simplify.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.1

Cancel the common factor.

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

Step 2.2.1.1.2

Rewrite the expression.

$- x - 3 \geq 0 \cdot 2$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $0$ by $2$ .

$- x - 3 \geq 0$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Add $3$ to both sides of the inequality.

$- x \geq 3$

Step 2.3.2

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

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Step 2.3.2.1

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

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

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.3.2.2.2

Divide $x$ by $1$ .

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

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Divide $3$ by $- 1$ .

$x \leq - 3$

Step 3

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

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

Step 4

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

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

Step 5

Solve the inequality.

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Step 5.1

Multiply both sides by $2$ .

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

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Cancel the common factor.

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

Step 5.2.1.1.2

Rewrite the expression.

$- x - 3 < 0 \cdot 2$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $0$ by $2$ .

$- x - 3 < 0$

Step 5.3

Solve for $x$ .

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Step 5.3.1

Add $3$ to both sides of the inequality.

$- x < 3$

Step 5.3.2

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

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Step 5.3.2.1

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

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

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.3.2.2.2

Divide $x$ by $1$ .

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

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Divide $3$ by $- 1$ .

$x > - 3$

Step 6

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

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

Step 7

Write as a piecewise.

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

Step 8

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

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Step 8.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8.2

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 8.3

Combine the numerators over the common denominator.

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

Step 8.4

Simplify the numerator.

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Step 8.4.1

Multiply $- 1$ by $2$ .

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

Step 8.4.2

Subtract $2$ from $- 3$ .

${\begin{cases} \frac{- x - 5}{2} & x \leq - 3 \\ - \frac{- x - 3}{2} - 1 & x > - 3 \end{cases}$

Step 8.5

Factor $- 1$ out of $- x$ .

${\begin{cases} \frac{- (x) - 5}{2} & x \leq - 3 \\ - \frac{- x - 3}{2} - 1 & x > - 3 \end{cases}$

Step 8.6

Rewrite $- 5$ as $- 1 (5)$ .

${\begin{cases} \frac{- (x) - 1 (5)}{2} & x \leq - 3 \\ - \frac{- x - 3}{2} - 1 & x > - 3 \end{cases}$

Step 8.7

Factor $- 1$ out of $- (x) - 1 (5)$ .

${\begin{cases} \frac{- (x + 5)}{2} & x \leq - 3 \\ - \frac{- x - 3}{2} - 1 & x > - 3 \end{cases}$

Step 8.8

Rewrite $- (x + 5)$ as $- 1 (x + 5)$ .

${\begin{cases} \frac{- 1 (x + 5)}{2} & x \leq - 3 \\ - \frac{- x - 3}{2} - 1 & x > - 3 \end{cases}$

Step 8.9

Move the negative in front of the fraction.

${\begin{cases} - \frac{x + 5}{2} & x \leq - 3 \\ - \frac{- x - 3}{2} - 1 & x > - 3 \end{cases}$

Step 9

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

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Step 9.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${\begin{cases} - \frac{x + 5}{2} & x \leq - 3 \\ - \frac{- x - 3}{2} - 1 \cdot \frac{2}{2} & x > - 3 \end{cases}$

Step 9.2

Combine $- 1$ and $\frac{2}{2}$ .

${\begin{cases} - \frac{x + 5}{2} & x \leq - 3 \\ - \frac{- x - 3}{2} + \frac{- 1 \cdot 2}{2} & x > - 3 \end{cases}$

Step 9.3

Combine the numerators over the common denominator.

${\begin{cases} - \frac{x + 5}{2} & x \leq - 3 \\ \frac{- (- x - 3) - 1 \cdot 2}{2} & x > - 3 \end{cases}$

Step 9.4

Simplify the numerator.

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Step 9.4.1

Factor $- 1$ out of $- (- x - 3) - 1 \cdot 2$ .

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Step 9.4.1.1

Factor $- 1$ out of $- 1 \cdot 2$ .

${\begin{cases} - \frac{x + 5}{2} & x \leq - 3 \\ \frac{- (- x - 3) - 1 (2)}{2} & x > - 3 \end{cases}$

Step 9.4.1.2

Factor $- 1$ out of $- (- x - 3) - 1 (2)$ .

${\begin{cases} - \frac{x + 5}{2} & x \leq - 3 \\ \frac{- (- x - 3 + 2)}{2} & x > - 3 \end{cases}$

Step 9.4.2

Add $- 3$ and $2$ .

${\begin{cases} - \frac{x + 5}{2} & x \leq - 3 \\ \frac{- (- x - 1)}{2} & x > - 3 \end{cases}$

Step 9.5

Move the negative in front of the fraction.

${\begin{cases} - \frac{x + 5}{2} & x \leq - 3 \\ - \frac{- x - 1}{2} & x > - 3 \end{cases}$

Step 9.6

Factor $- 1$ out of $- x$ .

${\begin{cases} - \frac{x + 5}{2} & x \leq - 3 \\ - \frac{- (x) - 1}{2} & x > - 3 \end{cases}$

Step 9.7

Rewrite $- 1$ as $- 1 (1)$ .

${\begin{cases} - \frac{x + 5}{2} & x \leq - 3 \\ - \frac{- (x) - 1 (1)}{2} & x > - 3 \end{cases}$

Step 9.8

Factor $- 1$ out of $- (x) - 1 (1)$ .

${\begin{cases} - \frac{x + 5}{2} & x \leq - 3 \\ - \frac{- (x + 1)}{2} & x > - 3 \end{cases}$

Step 9.9

Rewrite $- (x + 1)$ as $- 1 (x + 1)$ .

${\begin{cases} - \frac{x + 5}{2} & x \leq - 3 \\ - \frac{- 1 (x + 1)}{2} & x > - 3 \end{cases}$

Step 9.10

Move the negative in front of the fraction.

${\begin{cases} - \frac{x + 5}{2} & x \leq - 3 \\ - - \frac{x + 1}{2} & x > - 3 \end{cases}$

Step 9.11

Multiply $- 1$ by $- 1$ .

${\begin{cases} - \frac{x + 5}{2} & x \leq - 3 \\ 1 \frac{x + 1}{2} & x > - 3 \end{cases}$

Step 9.12

Multiply $\frac{x + 1}{2}$ by $1$ .

${\begin{cases} - \frac{x + 5}{2} & x \leq - 3 \\ \frac{x + 1}{2} & x > - 3 \end{cases}$