Solve for x |(3x)/8-5|>3/4

$| \frac{3 x}{8} - 5 | > \frac{3}{4}$

Write $| \frac{3 x}{8} - 5 | > \frac{3}{4}$ as a piecewise.

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

$\frac{3 x}{8} - 5 \geq 0$

Solve the inequality.

Add $5$ to both sides of the inequality.

$\frac{3 x}{8} \geq 5$

Multiply both sides by $8$ .

$\frac{3 x}{8} \cdot 8 \geq 5 \cdot 8$

Simplify.

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{3 x}{8} \cdot 8 \geq 5 \cdot 8$

Rewrite the expression.

$3 x \geq 5 \cdot 8$

Simplify the right side.

Multiply $5$ by $8$ .

$3 x \geq 40$

Divide each term in $3 x \geq 40$ by $3$ and simplify.

Divide each term in $3 x \geq 40$ by $3$ .

$\frac{3 x}{3} \geq \frac{40}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{40}{3}$

Divide $x$ by $1$ .

$x \geq \frac{40}{3}$

In the piece where $\frac{3 x}{8} - 5$ is non-negative, remove the absolute value.

$\frac{3 x}{8} - 5 > \frac{3}{4}$

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

$\frac{3 x}{8} - 5 < 0$

Solve the inequality.

Add $5$ to both sides of the inequality.

$\frac{3 x}{8} < 5$

Multiply both sides by $8$ .

$\frac{3 x}{8} \cdot 8 < 5 \cdot 8$

Simplify.

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{3 x}{8} \cdot 8 < 5 \cdot 8$

Rewrite the expression.

$3 x < 5 \cdot 8$

Simplify the right side.

Multiply $5$ by $8$ .

$3 x < 40$

Divide each term in $3 x < 40$ by $3$ and simplify.

Divide each term in $3 x < 40$ by $3$ .

$\frac{3 x}{3} < \frac{40}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{40}{3}$

Divide $x$ by $1$ .

$x < \frac{40}{3}$

In the piece where $\frac{3 x}{8} - 5$ is negative, remove the absolute value and multiply by $- 1$ .

$- (\frac{3 x}{8} - 5) > \frac{3}{4}$

Write as a piecewise.

${\begin{cases} \frac{3 x}{8} - 5 > \frac{3}{4} & x \geq \frac{40}{3} \\ - (\frac{3 x}{8} - 5) > \frac{3}{4} & x < \frac{40}{3} \end{cases}$

Simplify $- (\frac{3 x}{8} - 5) > \frac{3}{4}$ .

Apply the distributive property.

${\begin{cases} \frac{3 x}{8} - 5 > \frac{3}{4} & x \geq \frac{40}{3} \\ - \frac{3 x}{8} - - 5 > \frac{3}{4} & x < \frac{40}{3} \end{cases}$

Multiply $- 1$ by $- 5$ .

${\begin{cases} \frac{3 x}{8} - 5 > \frac{3}{4} & x \geq \frac{40}{3} \\ - \frac{3 x}{8} + 5 > \frac{3}{4} & x < \frac{40}{3} \end{cases}$

Solve $\frac{3 x}{8} - 5 > \frac{3}{4}$ for $x$ .

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

Add $5$ to both sides of the inequality.

$\frac{3 x}{8} > \frac{3}{4} + 5$

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3 x}{8} > \frac{3}{4} + 5 \cdot \frac{4}{4}$

Combine $5$ and $\frac{4}{4}$ .

$\frac{3 x}{8} > \frac{3}{4} + \frac{5 \cdot 4}{4}$

Combine the numerators over the common denominator.

$\frac{3 x}{8} > \frac{3 + 5 \cdot 4}{4}$

Simplify the numerator.

Multiply $5$ by $4$ .

$\frac{3 x}{8} > \frac{3 + 20}{4}$

Add $3$ and $20$ .

$\frac{3 x}{8} > \frac{23}{4}$

Multiply both sides by $8$ .

$\frac{3 x}{8} \cdot 8 > \frac{23}{4} \cdot 8$

Simplify.

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{3 x}{8} \cdot 8 > \frac{23}{4} \cdot 8$

Rewrite the expression.

$3 x > \frac{23}{4} \cdot 8$

Simplify the right side.

Simplify $\frac{23}{4} \cdot 8$ .

Cancel the common factor of $4$ .

Factor $4$ out of $8$ .

$3 x > \frac{23}{4} \cdot (4 (2))$

Cancel the common factor.

$3 x > \frac{23}{4} \cdot (4 \cdot 2)$

Rewrite the expression.

$3 x > 23 \cdot 2$

Multiply $23$ by $2$ .

$3 x > 46$

Divide each term in $3 x > 46$ by $3$ and simplify.

Divide each term in $3 x > 46$ by $3$ .

$\frac{3 x}{3} > \frac{46}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} > \frac{46}{3}$

Divide $x$ by $1$ .

$x > \frac{46}{3}$

Solve $- \frac{3 x}{8} + 5 > \frac{3}{4}$ for $x$ .

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

Subtract $5$ from both sides of the inequality.

$- \frac{3 x}{8} > \frac{3}{4} - 5$

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{3 x}{8} > \frac{3}{4} - 5 \cdot \frac{4}{4}$

Combine $- 5$ and $\frac{4}{4}$ .

$- \frac{3 x}{8} > \frac{3}{4} + \frac{- 5 \cdot 4}{4}$

Combine the numerators over the common denominator.

$- \frac{3 x}{8} > \frac{3 - 5 \cdot 4}{4}$

Simplify the numerator.

Multiply $- 5$ by $4$ .

$- \frac{3 x}{8} > \frac{3 - 20}{4}$

Subtract $20$ from $3$ .

$- \frac{3 x}{8} > \frac{- 17}{4}$

Move the negative in front of the fraction.

$- \frac{3 x}{8} > - \frac{17}{4}$

Divide each term in $- \frac{3 x}{8} > - \frac{17}{4}$ by $- 1$ and simplify.

Divide each term in $- \frac{3 x}{8} > - \frac{17}{4}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{3 x}{8}}{- 1} < \frac{- \frac{17}{4}}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{3 x}{8}}{1} < \frac{- \frac{17}{4}}{- 1}$

Divide $\frac{3 x}{8}$ by $1$ .

$\frac{3 x}{8} < \frac{- \frac{17}{4}}{- 1}$

Simplify the right side.

Dividing two negative values results in a positive value.

$\frac{3 x}{8} < \frac{\frac{17}{4}}{1}$

Divide $\frac{17}{4}$ by $1$ .

$\frac{3 x}{8} < \frac{17}{4}$

Multiply both sides by $8$ .

$\frac{3 x}{8} \cdot 8 < \frac{17}{4} \cdot 8$

Simplify.

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{3 x}{8} \cdot 8 < \frac{17}{4} \cdot 8$

Rewrite the expression.

$3 x < \frac{17}{4} \cdot 8$

Simplify the right side.

Simplify $\frac{17}{4} \cdot 8$ .

Cancel the common factor of $4$ .

Factor $4$ out of $8$ .

$3 x < \frac{17}{4} \cdot (4 (2))$

Cancel the common factor.

$3 x < \frac{17}{4} \cdot (4 \cdot 2)$

Rewrite the expression.

$3 x < 17 \cdot 2$

Multiply $17$ by $2$ .

$3 x < 34$

Divide each term in $3 x < 34$ by $3$ and simplify.

Divide each term in $3 x < 34$ by $3$ .

$\frac{3 x}{3} < \frac{34}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{34}{3}$

Divide $x$ by $1$ .

$x < \frac{34}{3}$

Find the union of the solutions.

$x < \frac{34}{3}$ or $x > \frac{46}{3}$

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{34}{3} or x > \frac{46}{3}$

Interval Notation:

$(- \infty, \frac{34}{3}) \cup (\frac{46}{3}, \infty)$

Solve for x |(3x)/8-5|>3/4

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