Pre-Algebra Examples

$| \frac{4}{7} x - 4 | > 12$

Step 1

Write $| \frac{4}{7} x - 4 | > 12$ 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.

$\frac{4}{7} x - 4 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Combine $\frac{4}{7}$ and $x$ .

$\frac{4 x}{7} - 4 \geq 0$

Step 1.2.2

Add $4$ to both sides of the inequality.

$\frac{4 x}{7} \geq 4$

Step 1.2.3

Multiply both sides by $7$ .

$\frac{4 x}{7} \cdot 7 \geq 4 \cdot 7$

Step 1.2.4

Simplify.

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Step 1.2.4.1

Simplify the left side.

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Step 1.2.4.1.1

Cancel the common factor of $7$ .

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Step 1.2.4.1.1.1

Cancel the common factor.

$\frac{4 x}{7} \cdot 7 \geq 4 \cdot 7$

Step 1.2.4.1.1.2

Rewrite the expression.

$4 x \geq 4 \cdot 7$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply $4$ by $7$ .

$4 x \geq 28$

Step 1.2.5

Divide each term in $4 x \geq 28$ by $4$ and simplify.

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Step 1.2.5.1

Divide each term in $4 x \geq 28$ by $4$ .

$\frac{4 x}{4} \geq \frac{28}{4}$

Step 1.2.5.2

Simplify the left side.

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Step 1.2.5.2.1

Cancel the common factor of $4$ .

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Step 1.2.5.2.1.1

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{28}{4}$

Step 1.2.5.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{28}{4}$

Step 1.2.5.3

Simplify the right side.

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Step 1.2.5.3.1

Divide $28$ by $4$ .

$x \geq 7$

Step 1.3

In the piece where $\frac{4}{7} x - 4$ is non-negative, remove the absolute value.

$\frac{4}{7} x - 4 > 12$

Step 1.4

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

$\frac{4}{7} x - 4 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Combine $\frac{4}{7}$ and $x$ .

$\frac{4 x}{7} - 4 < 0$

Step 1.5.2

Add $4$ to both sides of the inequality.

$\frac{4 x}{7} < 4$

Step 1.5.3

Multiply both sides by $7$ .

$\frac{4 x}{7} \cdot 7 < 4 \cdot 7$

Step 1.5.4

Simplify.

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Step 1.5.4.1

Simplify the left side.

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Step 1.5.4.1.1

Cancel the common factor of $7$ .

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Step 1.5.4.1.1.1

Cancel the common factor.

$\frac{4 x}{7} \cdot 7 < 4 \cdot 7$

Step 1.5.4.1.1.2

Rewrite the expression.

$4 x < 4 \cdot 7$

Step 1.5.4.2

Simplify the right side.

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Step 1.5.4.2.1

Multiply $4$ by $7$ .

$4 x < 28$

Step 1.5.5

Divide each term in $4 x < 28$ by $4$ and simplify.

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Step 1.5.5.1

Divide each term in $4 x < 28$ by $4$ .

$\frac{4 x}{4} < \frac{28}{4}$

Step 1.5.5.2

Simplify the left side.

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Step 1.5.5.2.1

Cancel the common factor of $4$ .

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Step 1.5.5.2.1.1

Cancel the common factor.

$\frac{4 x}{4} < \frac{28}{4}$

Step 1.5.5.2.1.2

Divide $x$ by $1$ .

$x < \frac{28}{4}$

Step 1.5.5.3

Simplify the right side.

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Step 1.5.5.3.1

Divide $28$ by $4$ .

$x < 7$

Step 1.6

In the piece where $\frac{4}{7} x - 4$ is negative, remove the absolute value and multiply by $- 1$ .

$- (\frac{4}{7} x - 4) > 12$

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{4}{7} x - 4 > 12 & x \geq 7 \\ - (\frac{4}{7} x - 4) > 12 & x < 7 \end{cases}$

Step 1.8

Combine $\frac{4}{7}$ and $x$ .

${\begin{cases} \frac{4 x}{7} - 4 > 12 & x \geq 7 \\ - (\frac{4}{7} x - 4) > 12 & x < 7 \end{cases}$

Step 1.9

Simplify $- (\frac{4}{7} x - 4) > 12$ .

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Step 1.9.1

Combine $\frac{4}{7}$ and $x$ .

${\begin{cases} \frac{4 x}{7} - 4 > 12 & x \geq 7 \\ - (\frac{4 x}{7} - 4) > 12 & x < 7 \end{cases}$

Step 1.9.2

Apply the distributive property.

${\begin{cases} \frac{4 x}{7} - 4 > 12 & x \geq 7 \\ - \frac{4 x}{7} - - 4 > 12 & x < 7 \end{cases}$

Step 1.9.3

Multiply $- 1$ by $- 4$ .

${\begin{cases} \frac{4 x}{7} - 4 > 12 & x \geq 7 \\ - \frac{4 x}{7} + 4 > 12 & x < 7 \end{cases}$

Step 2

Solve $\frac{4 x}{7} - 4 > 12$ for $x$ .

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Step 2.1

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

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Step 2.1.1

Add $4$ to both sides of the inequality.

$\frac{4 x}{7} > 12 + 4$

Step 2.1.2

Add $12$ and $4$ .

$\frac{4 x}{7} > 16$

Step 2.2

Multiply both sides by $7$ .

$\frac{4 x}{7} \cdot 7 > 16 \cdot 7$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Cancel the common factor of $7$ .

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Step 2.3.1.1.1

Cancel the common factor.

$\frac{4 x}{7} \cdot 7 > 16 \cdot 7$

Step 2.3.1.1.2

Rewrite the expression.

$4 x > 16 \cdot 7$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $16$ by $7$ .

$4 x > 112$

Step 2.4

Divide each term in $4 x > 112$ by $4$ and simplify.

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Step 2.4.1

Divide each term in $4 x > 112$ by $4$ .

$\frac{4 x}{4} > \frac{112}{4}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $4$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} > \frac{112}{4}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x > \frac{112}{4}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $112$ by $4$ .

$x > 28$

Step 3

Solve $- \frac{4 x}{7} + 4 > 12$ for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $4$ from both sides of the inequality.

$- \frac{4 x}{7} > 12 - 4$

Step 3.1.2

Subtract $4$ from $12$ .

$- \frac{4 x}{7} > 8$

Step 3.2

Divide each term in $- \frac{4 x}{7} > 8$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- \frac{4 x}{7} > 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{4 x}{7}}{- 1} < \frac{8}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{4 x}{7}}{1} < \frac{8}{- 1}$

Step 3.2.2.2

Divide $\frac{4 x}{7}$ by $1$ .

$\frac{4 x}{7} < \frac{8}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $8$ by $- 1$ .

$\frac{4 x}{7} < - 8$

Step 3.3

Multiply both sides by $7$ .

$\frac{4 x}{7} \cdot 7 < - 8 \cdot 7$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Cancel the common factor of $7$ .

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Step 3.4.1.1.1

Cancel the common factor.

$\frac{4 x}{7} \cdot 7 < - 8 \cdot 7$

Step 3.4.1.1.2

Rewrite the expression.

$4 x < - 8 \cdot 7$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Multiply $- 8$ by $7$ .

$4 x < - 56$

Step 3.5

Divide each term in $4 x < - 56$ by $4$ and simplify.

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Step 3.5.1

Divide each term in $4 x < - 56$ by $4$ .

$\frac{4 x}{4} < \frac{- 56}{4}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $4$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{4 x}{4} < \frac{- 56}{4}$

Step 3.5.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 56}{4}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Divide $- 56$ by $4$ .

$x < - 14$

Step 4

Find the union of the solutions.

$x < - 14$ or $x > 28$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < - 14 or x > 28$

Interval Notation:

$(- \infty, - 14) \cup (28, \infty)$

Step 6