Pre-Algebra Examples

$| \frac{x}{3} - \frac{x}{4} | - 1 < 0$

Step 1

Write $| \frac{x}{3} - \frac{x}{4} | - 1 < 0$ as a piecewise.

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Step 1.1

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

$\frac{x}{3} - \frac{x}{4} \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Simplify $\frac{x}{3} - \frac{x}{4}$ .

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Step 1.2.1.1

To write $\frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x}{3} \cdot \frac{4}{4} - \frac{x}{4} \geq 0$

Step 1.2.1.2

To write $- \frac{x}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{3} \cdot \frac{4}{4} - \frac{x}{4} \cdot \frac{3}{3} \geq 0$

Step 1.2.1.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1.3.1

Multiply $\frac{x}{3}$ by $\frac{4}{4}$ .

$\frac{x \cdot 4}{3 \cdot 4} - \frac{x}{4} \cdot \frac{3}{3} \geq 0$

Step 1.2.1.3.2

Multiply $3$ by $4$ .

$\frac{x \cdot 4}{12} - \frac{x}{4} \cdot \frac{3}{3} \geq 0$

Step 1.2.1.3.3

Multiply $\frac{x}{4}$ by $\frac{3}{3}$ .

$\frac{x \cdot 4}{12} - \frac{x \cdot 3}{4 \cdot 3} \geq 0$

Step 1.2.1.3.4

Multiply $4$ by $3$ .

$\frac{x \cdot 4}{12} - \frac{x \cdot 3}{12} \geq 0$

Step 1.2.1.4

Combine the numerators over the common denominator.

$\frac{x \cdot 4 - x \cdot 3}{12} \geq 0$

Step 1.2.1.5

Simplify the numerator.

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Step 1.2.1.5.1

Factor $x$ out of $x \cdot 4 - x \cdot 3$ .

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Step 1.2.1.5.1.1

Factor $x$ out of $x \cdot 4$ .

$\frac{x (4) - x \cdot 3}{12} \geq 0$

Step 1.2.1.5.1.2

Factor $x$ out of $- x \cdot 3$ .

$\frac{x (4) + x (- 1 \cdot 3)}{12} \geq 0$

Step 1.2.1.5.1.3

Factor $x$ out of $x (4) + x (- 1 \cdot 3)$ .

$\frac{x (4 - 1 \cdot 3)}{12} \geq 0$

Step 1.2.1.5.2

Multiply $- 1$ by $3$ .

$\frac{x (4 - 3)}{12} \geq 0$

Step 1.2.1.5.3

Subtract $3$ from $4$ .

$\frac{x \cdot 1}{12} \geq 0$

Step 1.2.1.6

Multiply $x$ by $1$ .

$\frac{x}{12} \geq 0$

Step 1.2.2

Multiply both sides by $12$ .

$\frac{x}{12} \cdot 12 \geq 0 \cdot 12$

Step 1.2.3

Simplify.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Cancel the common factor of $12$ .

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Step 1.2.3.1.1.1

Cancel the common factor.

$\frac{x}{12} \cdot 12 \geq 0 \cdot 12$

Step 1.2.3.1.1.2

Rewrite the expression.

$x \geq 0 \cdot 12$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $0$ by $12$ .

$x \geq 0$

Step 1.3

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

$\frac{x}{3} - \frac{x}{4} - 1 < 0$

Step 1.4

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

$\frac{x}{3} - \frac{x}{4} < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Simplify $\frac{x}{3} - \frac{x}{4}$ .

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Step 1.5.1.1

To write $\frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x}{3} \cdot \frac{4}{4} - \frac{x}{4} < 0$

Step 1.5.1.2

To write $- \frac{x}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{3} \cdot \frac{4}{4} - \frac{x}{4} \cdot \frac{3}{3} < 0$

Step 1.5.1.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.1.3.1

Multiply $\frac{x}{3}$ by $\frac{4}{4}$ .

$\frac{x \cdot 4}{3 \cdot 4} - \frac{x}{4} \cdot \frac{3}{3} < 0$

Step 1.5.1.3.2

Multiply $3$ by $4$ .

$\frac{x \cdot 4}{12} - \frac{x}{4} \cdot \frac{3}{3} < 0$

Step 1.5.1.3.3

Multiply $\frac{x}{4}$ by $\frac{3}{3}$ .

$\frac{x \cdot 4}{12} - \frac{x \cdot 3}{4 \cdot 3} < 0$

Step 1.5.1.3.4

Multiply $4$ by $3$ .

$\frac{x \cdot 4}{12} - \frac{x \cdot 3}{12} < 0$

Step 1.5.1.4

Combine the numerators over the common denominator.

$\frac{x \cdot 4 - x \cdot 3}{12} < 0$

Step 1.5.1.5

Simplify the numerator.

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Step 1.5.1.5.1

Factor $x$ out of $x \cdot 4 - x \cdot 3$ .

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Step 1.5.1.5.1.1

Factor $x$ out of $x \cdot 4$ .

$\frac{x (4) - x \cdot 3}{12} < 0$

Step 1.5.1.5.1.2

Factor $x$ out of $- x \cdot 3$ .

$\frac{x (4) + x (- 1 \cdot 3)}{12} < 0$

Step 1.5.1.5.1.3

Factor $x$ out of $x (4) + x (- 1 \cdot 3)$ .

$\frac{x (4 - 1 \cdot 3)}{12} < 0$

Step 1.5.1.5.2

Multiply $- 1$ by $3$ .

$\frac{x (4 - 3)}{12} < 0$

Step 1.5.1.5.3

Subtract $3$ from $4$ .

$\frac{x \cdot 1}{12} < 0$

Step 1.5.1.6

Multiply $x$ by $1$ .

$\frac{x}{12} < 0$

Step 1.5.2

Multiply both sides by $12$ .

$\frac{x}{12} \cdot 12 < 0 \cdot 12$

Step 1.5.3

Simplify.

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Step 1.5.3.1

Simplify the left side.

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Step 1.5.3.1.1

Cancel the common factor of $12$ .

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Step 1.5.3.1.1.1

Cancel the common factor.

$\frac{x}{12} \cdot 12 < 0 \cdot 12$

Step 1.5.3.1.1.2

Rewrite the expression.

$x < 0 \cdot 12$

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Multiply $0$ by $12$ .

$x < 0$

Step 1.6

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

$- (\frac{x}{3} - \frac{x}{4}) - 1 < 0$

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{x}{3} - \frac{x}{4} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} - \frac{x}{4}) - 1 < 0 & x < 0 \end{cases}$

Step 1.8

Simplify $\frac{x}{3} - \frac{x}{4} - 1 < 0$ .

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Step 1.8.1

To write $\frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

${\begin{cases} \frac{x}{3} \cdot \frac{4}{4} - \frac{x}{4} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} - \frac{x}{4}) - 1 < 0 & x < 0 \end{cases}$

Step 1.8.2

To write $- \frac{x}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${\begin{cases} \frac{x}{3} \cdot \frac{4}{4} - \frac{x}{4} \cdot \frac{3}{3} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} - \frac{x}{4}) - 1 < 0 & x < 0 \end{cases}$

Step 1.8.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.8.3.1

Multiply $\frac{x}{3}$ by $\frac{4}{4}$ .

${\begin{cases} \frac{x \cdot 4}{3 \cdot 4} - \frac{x}{4} \cdot \frac{3}{3} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} - \frac{x}{4}) - 1 < 0 & x < 0 \end{cases}$

Step 1.8.3.2

Multiply $3$ by $4$ .

${\begin{cases} \frac{x \cdot 4}{12} - \frac{x}{4} \cdot \frac{3}{3} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} - \frac{x}{4}) - 1 < 0 & x < 0 \end{cases}$

Step 1.8.3.3

Multiply $\frac{x}{4}$ by $\frac{3}{3}$ .

${\begin{cases} \frac{x \cdot 4}{12} - \frac{x \cdot 3}{4 \cdot 3} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} - \frac{x}{4}) - 1 < 0 & x < 0 \end{cases}$

Step 1.8.3.4

Multiply $4$ by $3$ .

${\begin{cases} \frac{x \cdot 4}{12} - \frac{x \cdot 3}{12} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} - \frac{x}{4}) - 1 < 0 & x < 0 \end{cases}$

Step 1.8.4

Combine the numerators over the common denominator.

${\begin{cases} \frac{x \cdot 4 - x \cdot 3}{12} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} - \frac{x}{4}) - 1 < 0 & x < 0 \end{cases}$

Step 1.8.5

Simplify each term.

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Step 1.8.5.1

Simplify the numerator.

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Step 1.8.5.1.1

Factor $x$ out of $x \cdot 4 - x \cdot 3$ .

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Step 1.8.5.1.1.1

Factor $x$ out of $x \cdot 4$ .

${\begin{cases} \frac{x (4) - x \cdot 3}{12} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} - \frac{x}{4}) - 1 < 0 & x < 0 \end{cases}$

Step 1.8.5.1.1.2

Factor $x$ out of $- x \cdot 3$ .

${\begin{cases} \frac{x (4) + x (- 1 \cdot 3)}{12} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} - \frac{x}{4}) - 1 < 0 & x < 0 \end{cases}$

Step 1.8.5.1.1.3

Factor $x$ out of $x (4) + x (- 1 \cdot 3)$ .

${\begin{cases} \frac{x (4 - 1 \cdot 3)}{12} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} - \frac{x}{4}) - 1 < 0 & x < 0 \end{cases}$

Step 1.8.5.1.2

Multiply $- 1$ by $3$ .

${\begin{cases} \frac{x (4 - 3)}{12} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} - \frac{x}{4}) - 1 < 0 & x < 0 \end{cases}$

Step 1.8.5.1.3

Subtract $3$ from $4$ .

${\begin{cases} \frac{x \cdot 1}{12} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} - \frac{x}{4}) - 1 < 0 & x < 0 \end{cases}$

Step 1.8.5.2

Multiply $x$ by $1$ .

${\begin{cases} \frac{x}{12} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} - \frac{x}{4}) - 1 < 0 & x < 0 \end{cases}$

Step 1.9

Simplify each term.

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Step 1.9.1

To write $\frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

${\begin{cases} \frac{x}{12} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} \cdot \frac{4}{4} - \frac{x}{4}) - 1 < 0 & x < 0 \end{cases}$

Step 1.9.2

To write $- \frac{x}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${\begin{cases} \frac{x}{12} - 1 < 0 & x \geq 0 \\ - (\frac{x}{3} \cdot \frac{4}{4} - \frac{x}{4} \cdot \frac{3}{3}) - 1 < 0 & x < 0 \end{cases}$

Step 1.9.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.9.3.1

Multiply $\frac{x}{3}$ by $\frac{4}{4}$ .

${\begin{cases} \frac{x}{12} - 1 < 0 & x \geq 0 \\ - (\frac{x \cdot 4}{3 \cdot 4} - \frac{x}{4} \cdot \frac{3}{3}) - 1 < 0 & x < 0 \end{cases}$

Step 1.9.3.2

Multiply $3$ by $4$ .

${\begin{cases} \frac{x}{12} - 1 < 0 & x \geq 0 \\ - (\frac{x \cdot 4}{12} - \frac{x}{4} \cdot \frac{3}{3}) - 1 < 0 & x < 0 \end{cases}$

Step 1.9.3.3

Multiply $\frac{x}{4}$ by $\frac{3}{3}$ .

${\begin{cases} \frac{x}{12} - 1 < 0 & x \geq 0 \\ - (\frac{x \cdot 4}{12} - \frac{x \cdot 3}{4 \cdot 3}) - 1 < 0 & x < 0 \end{cases}$

Step 1.9.3.4

Multiply $4$ by $3$ .

${\begin{cases} \frac{x}{12} - 1 < 0 & x \geq 0 \\ - (\frac{x \cdot 4}{12} - \frac{x \cdot 3}{12}) - 1 < 0 & x < 0 \end{cases}$

Step 1.9.4

Combine the numerators over the common denominator.

${\begin{cases} \frac{x}{12} - 1 < 0 & x \geq 0 \\ - \frac{x \cdot 4 - x \cdot 3}{12} - 1 < 0 & x < 0 \end{cases}$

Step 1.9.5

Simplify the numerator.

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Step 1.9.5.1

Factor $x$ out of $x \cdot 4 - x \cdot 3$ .

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Step 1.9.5.1.1

Factor $x$ out of $x \cdot 4$ .

${\begin{cases} \frac{x}{12} - 1 < 0 & x \geq 0 \\ - \frac{x (4) - x \cdot 3}{12} - 1 < 0 & x < 0 \end{cases}$

Step 1.9.5.1.2

Factor $x$ out of $- x \cdot 3$ .

${\begin{cases} \frac{x}{12} - 1 < 0 & x \geq 0 \\ - \frac{x (4) + x (- 1 \cdot 3)}{12} - 1 < 0 & x < 0 \end{cases}$

Step 1.9.5.1.3

Factor $x$ out of $x (4) + x (- 1 \cdot 3)$ .

${\begin{cases} \frac{x}{12} - 1 < 0 & x \geq 0 \\ - \frac{x (4 - 1 \cdot 3)}{12} - 1 < 0 & x < 0 \end{cases}$

Step 1.9.5.2

Multiply $- 1$ by $3$ .

${\begin{cases} \frac{x}{12} - 1 < 0 & x \geq 0 \\ - \frac{x (4 - 3)}{12} - 1 < 0 & x < 0 \end{cases}$

Step 1.9.5.3

Subtract $3$ from $4$ .

${\begin{cases} \frac{x}{12} - 1 < 0 & x \geq 0 \\ - \frac{x \cdot 1}{12} - 1 < 0 & x < 0 \end{cases}$

Step 1.9.6

Multiply $x$ by $1$ .

${\begin{cases} \frac{x}{12} - 1 < 0 & x \geq 0 \\ - \frac{x}{12} - 1 < 0 & x < 0 \end{cases}$

Step 2

Solve $\frac{x}{12} - 1 < 0$ when $x \geq 0$ .

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Step 2.1

Solve $\frac{x}{12} - 1 < 0$ for $x$ .

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Step 2.1.1

Add $1$ to both sides of the inequality.

$\frac{x}{12} < 1$

Step 2.1.2

Multiply both sides by $12$ .

$\frac{x}{12} \cdot 12 < 1 \cdot 12$

Step 2.1.3

Simplify.

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Step 2.1.3.1

Simplify the left side.

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Step 2.1.3.1.1

Cancel the common factor of $12$ .

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Step 2.1.3.1.1.1

Cancel the common factor.

$\frac{x}{12} \cdot 12 < 1 \cdot 12$

Step 2.1.3.1.1.2

Rewrite the expression.

$x < 1 \cdot 12$

Step 2.1.3.2

Simplify the right side.

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Step 2.1.3.2.1

Multiply $12$ by $1$ .

$x < 12$

Step 2.2

Find the intersection of $x < 12$ and $x \geq 0$ .

$0 \leq x < 12$

Step 3

Solve $- \frac{x}{12} - 1 < 0$ when $x < 0$ .

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Step 3.1

Solve $- \frac{x}{12} - 1 < 0$ for $x$ .

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Step 3.1.1

Add $1$ to both sides of the inequality.

$- \frac{x}{12} < 1$

Step 3.1.2

Divide each term in $- \frac{x}{12} < 1$ by $- 1$ and simplify.

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Step 3.1.2.1

Divide each term in $- \frac{x}{12} < 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{12}}{- 1} > \frac{1}{- 1}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{12}}{1} > \frac{1}{- 1}$

Step 3.1.2.2.2

Divide $\frac{x}{12}$ by $1$ .

$\frac{x}{12} > \frac{1}{- 1}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $1$ by $- 1$ .

$\frac{x}{12} > - 1$

Step 3.1.3

Multiply both sides by $12$ .

$\frac{x}{12} \cdot 12 > - 1 \cdot 12$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Simplify the left side.

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Step 3.1.4.1.1

Cancel the common factor of $12$ .

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Step 3.1.4.1.1.1

Cancel the common factor.

$\frac{x}{12} \cdot 12 > - 1 \cdot 12$

Step 3.1.4.1.1.2

Rewrite the expression.

$x > - 1 \cdot 12$

Step 3.1.4.2

Simplify the right side.

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Step 3.1.4.2.1

Multiply $- 1$ by $12$ .

$x > - 12$

Step 3.2

Find the intersection of $x > - 12$ and $x < 0$ .

$- 12 < x < 0$

Step 4

Find the union of the solutions.

$- 12 < x < 12$

Step 5