Solve the Inequality for x |3/4x-3|-8>=-5

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

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

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

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

Solve the inequality.

Combine $\frac{3}{4}$ and $x$ .

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

Add $3$ to both sides of the inequality.

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

Multiply both sides by $4$ .

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

Simplify.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$3 x \geq 3 \cdot 4$

Simplify the right side.

Multiply $3$ by $4$ .

$3 x \geq 12$

Divide each term in $3 x \geq 12$ by $3$ and simplify.

Divide each term in $3 x \geq 12$ by $3$ .

$\frac{3 x}{3} \geq \frac{12}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{12}{3}$

Divide $x$ by $1$ .

$x \geq \frac{12}{3}$

Simplify the right side.

Divide $12$ by $3$ .

$x \geq 4$

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

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

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

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

Solve the inequality.

Combine $\frac{3}{4}$ and $x$ .

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

Add $3$ to both sides of the inequality.

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

Multiply both sides by $4$ .

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

Simplify.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$3 x < 3 \cdot 4$

Simplify the right side.

Multiply $3$ by $4$ .

$3 x < 12$

Divide each term in $3 x < 12$ by $3$ and simplify.

Divide each term in $3 x < 12$ by $3$ .

$\frac{3 x}{3} < \frac{12}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{12}{3}$

Divide $x$ by $1$ .

$x < \frac{12}{3}$

Simplify the right side.

Divide $12$ by $3$ .

$x < 4$

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

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

Write as a piecewise.

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

Simplify $\frac{3}{4} x - 3 - 8 \geq - 5$ .

Combine $\frac{3}{4}$ and $x$ .

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

Subtract $8$ from $- 3$ .

${\begin{cases} \frac{3 x}{4} - 11 \geq - 5 & x \geq 4 \\ - (\frac{3}{4} x - 3) - 8 \geq - 5 & x < 4 \end{cases}$

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

Simplify each term.

Combine $\frac{3}{4}$ and $x$ .

${\begin{cases} \frac{3 x}{4} - 11 \geq - 5 & x \geq 4 \\ - (\frac{3 x}{4} - 3) - 8 \geq - 5 & x < 4 \end{cases}$

Apply the distributive property.

${\begin{cases} \frac{3 x}{4} - 11 \geq - 5 & x \geq 4 \\ - \frac{3 x}{4} - - 3 - 8 \geq - 5 & x < 4 \end{cases}$

Multiply $- 1$ by $- 3$ .

${\begin{cases} \frac{3 x}{4} - 11 \geq - 5 & x \geq 4 \\ - \frac{3 x}{4} + 3 - 8 \geq - 5 & x < 4 \end{cases}$

Subtract $8$ from $3$ .

${\begin{cases} \frac{3 x}{4} - 11 \geq - 5 & x \geq 4 \\ - \frac{3 x}{4} - 5 \geq - 5 & x < 4 \end{cases}$

Solve $\frac{3 x}{4} - 11 \geq - 5$ for $x$ .

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

Add $11$ to both sides of the inequality.

$\frac{3 x}{4} \geq - 5 + 11$

Add $- 5$ and $11$ .

$\frac{3 x}{4} \geq 6$

Multiply both sides by $4$ .

$\frac{3 x}{4} \cdot 4 \geq 6 \cdot 4$

Simplify.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{3 x}{4} \cdot 4 \geq 6 \cdot 4$

Rewrite the expression.

$3 x \geq 6 \cdot 4$

Simplify the right side.

Multiply $6$ by $4$ .

$3 x \geq 24$

Divide each term in $3 x \geq 24$ by $3$ and simplify.

Divide each term in $3 x \geq 24$ by $3$ .

$\frac{3 x}{3} \geq \frac{24}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{24}{3}$

Divide $x$ by $1$ .

$x \geq \frac{24}{3}$

Simplify the right side.

Divide $24$ by $3$ .

$x \geq 8$

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

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

Add $5$ to both sides of the inequality.

$- \frac{3 x}{4} \geq - 5 + 5$

Add $- 5$ and $5$ .

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

Divide each term in $- \frac{3 x}{4} \geq 0$ by $- 1$ and simplify.

Divide each term in $- \frac{3 x}{4} \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $\frac{3 x}{4}$ by $1$ .

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

Simplify the right side.

Divide $0$ by $- 1$ .

$\frac{3 x}{4} \leq 0$

Multiply both sides by $4$ .

$\frac{3 x}{4} \cdot 4 \leq 0 \cdot 4$

Simplify.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{3 x}{4} \cdot 4 \leq 0 \cdot 4$

Rewrite the expression.

$3 x \leq 0 \cdot 4$

Simplify the right side.

Multiply $0$ by $4$ .

$3 x \leq 0$

Divide each term in $3 x \leq 0$ by $3$ and simplify.

Divide each term in $3 x \leq 0$ by $3$ .

$\frac{3 x}{3} \leq \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \leq \frac{0}{3}$

Divide $x$ by $1$ .

$x \leq \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x \leq 0$

Find the union of the solutions.

$x \leq 0$ or $x \geq 8$

The result can be shown in multiple forms.

Inequality Form:

$x \leq 0 or x \geq 8$

Interval Notation:

$(- \infty, 0] \cup [8, \infty)$

Solve the Inequality for x |3/4x-3|-8>=-5

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