Convert to Interval Notation |4/7x-4|>8

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

Step 1

Write $| \frac{4}{7} x - 4 | > 8$ as a piecewise.

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

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

Solve the inequality.

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

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

Add $4$ to both sides of the inequality.

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

Multiply both sides by $7$ .

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

Simplify.

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

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

Rewrite the expression.

$4 x \geq 4 \cdot 7$

Simplify the right side.

Multiply $4$ by $7$ .

$4 x \geq 28$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $28$ by $4$ .

$x \geq 7$

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

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

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

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

Solve the inequality.

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

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

Add $4$ to both sides of the inequality.

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

Multiply both sides by $7$ .

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

Simplify.

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

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

Rewrite the expression.

$4 x < 4 \cdot 7$

Simplify the right side.

Multiply $4$ by $7$ .

$4 x < 28$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $28$ by $4$ .

$x < 7$

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

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

Write as a piecewise.

${\begin{cases} \frac{4}{7} x - 4 > 8 & x \geq 7 \\ - (\frac{4}{7} x - 4) > 8 & x < 7 \end{cases}$

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

${\begin{cases} \frac{4 x}{7} - 4 > 8 & x \geq 7 \\ - (\frac{4}{7} x - 4) > 8 & x < 7 \end{cases}$

Simplify $- (\frac{4}{7} x - 4) > 8$ .

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

${\begin{cases} \frac{4 x}{7} - 4 > 8 & x \geq 7 \\ - (\frac{4 x}{7} - 4) > 8 & x < 7 \end{cases}$

Apply the distributive property.

${\begin{cases} \frac{4 x}{7} - 4 > 8 & x \geq 7 \\ - \frac{4 x}{7} - - 4 > 8 & x < 7 \end{cases}$

Multiply $- 1$ by $- 4$ .

${\begin{cases} \frac{4 x}{7} - 4 > 8 & x \geq 7 \\ - \frac{4 x}{7} + 4 > 8 & x < 7 \end{cases}$

Step 2

Solve $\frac{4 x}{7} - 4 > 8$ for $x$ .

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

Add $4$ to both sides of the inequality.

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

Add $8$ and $4$ .

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

Multiply both sides by $7$ .

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

Simplify.

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

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

Rewrite the expression.

$4 x > 12 \cdot 7$

Simplify the right side.

Multiply $12$ by $7$ .

$4 x > 84$

Divide each term in $4 x > 84$ by $4$ and simplify.

Divide each term in $4 x > 84$ by $4$ .

$\frac{4 x}{4} > \frac{84}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} > \frac{84}{4}$

Divide $x$ by $1$ .

$x > \frac{84}{4}$

Simplify the right side.

Divide $84$ by $4$ .

$x > 21$

Step 3

Solve $- \frac{4 x}{7} + 4 > 8$ for $x$ .

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

Subtract $4$ from both sides of the inequality.

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

Subtract $4$ from $8$ .

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

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

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

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

Simplify the right side.

Divide $4$ by $- 1$ .

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

Multiply both sides by $7$ .

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

Simplify.

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

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

Rewrite the expression.

$4 x < - 4 \cdot 7$

Simplify the right side.

Multiply $- 4$ by $7$ .

$4 x < - 28$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 28$ by $4$ .

$x < - 7$

Step 4

Find the union of the solutions.

$x < - 7$ or $x > 21$

Step 5

Convert the inequality to interval notation.

$(- \infty, - 7) \cup (21, \infty)$

Step 6

Convert to Interval Notation |4/7x-4|>8

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