Convert to Interval Notation |1-6x|-11<-8

$| 1 - 6 x | - 11 < - 8$

Step 1

Write $| 1 - 6 x | - 11 < - 8$ as a piecewise.

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

$1 - 6 x \geq 0$

Solve the inequality.

Subtract $1$ from both sides of the inequality.

$- 6 x \geq - 1$

Divide each term in $- 6 x \geq - 1$ by $- 6$ and simplify.

Divide each term in $- 6 x \geq - 1$ by $- 6$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 6 x}{- 6} \leq \frac{- 1}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 x}{- 6} \leq \frac{- 1}{- 6}$

Divide $x$ by $1$ .

$x \leq \frac{- 1}{- 6}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x \leq \frac{1}{6}$

In the piece where $1 - 6 x$ is non-negative, remove the absolute value.

$1 - 6 x - 11 < - 8$

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

$1 - 6 x < 0$

Solve the inequality.

Subtract $1$ from both sides of the inequality.

$- 6 x < - 1$

Divide each term in $- 6 x < - 1$ by $- 6$ and simplify.

Divide each term in $- 6 x < - 1$ by $- 6$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 6 x}{- 6} > \frac{- 1}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 x}{- 6} > \frac{- 1}{- 6}$

Divide $x$ by $1$ .

$x > \frac{- 1}{- 6}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x > \frac{1}{6}$

In the piece where $1 - 6 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (1 - 6 x) - 11 < - 8$

Write as a piecewise.

${\begin{cases} 1 - 6 x - 11 < - 8 & x \leq \frac{1}{6} \\ - (1 - 6 x) - 11 < - 8 & x > \frac{1}{6} \end{cases}$

Subtract $11$ from $1$ .

${\begin{cases} - 6 x - 10 < - 8 & x \leq \frac{1}{6} \\ - (1 - 6 x) - 11 < - 8 & x > \frac{1}{6} \end{cases}$

Simplify $- (1 - 6 x) - 11 < - 8$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 6 x - 10 < - 8 & x \leq \frac{1}{6} \\ - 1 \cdot 1 - (- 6 x) - 11 < - 8 & x > \frac{1}{6} \end{cases}$

Multiply $- 1$ by $1$ .

${\begin{cases} - 6 x - 10 < - 8 & x \leq \frac{1}{6} \\ - 1 - (- 6 x) - 11 < - 8 & x > \frac{1}{6} \end{cases}$

Multiply $- 6$ by $- 1$ .

${\begin{cases} - 6 x - 10 < - 8 & x \leq \frac{1}{6} \\ - 1 + 6 x - 11 < - 8 & x > \frac{1}{6} \end{cases}$

Subtract $11$ from $- 1$ .

${\begin{cases} - 6 x - 10 < - 8 & x \leq \frac{1}{6} \\ 6 x - 12 < - 8 & x > \frac{1}{6} \end{cases}$

Step 2

Solve $- 6 x - 10 < - 8$ when $x \leq \frac{1}{6}$ .

Solve $- 6 x - 10 < - 8$ for $x$ .

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

Add $10$ to both sides of the inequality.

$- 6 x < - 8 + 10$

Add $- 8$ and $10$ .

$- 6 x < 2$

Divide each term in $- 6 x < 2$ by $- 6$ and simplify.

Divide each term in $- 6 x < 2$ by $- 6$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 6 x}{- 6} > \frac{2}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 x}{- 6} > \frac{2}{- 6}$

Divide $x$ by $1$ .

$x > \frac{2}{- 6}$

Simplify the right side.

Cancel the common factor of $2$ and $- 6$ .

Factor $2$ out of $2$ .

$x > \frac{2 (1)}{- 6}$

Cancel the common factors.

Factor $2$ out of $- 6$ .

$x > \frac{2 \cdot 1}{2 \cdot - 3}$

Cancel the common factor.

$x > \frac{2 \cdot 1}{2 \cdot - 3}$

Rewrite the expression.

$x > \frac{1}{- 3}$

Move the negative in front of the fraction.

$x > - \frac{1}{3}$

Find the intersection of $x > - \frac{1}{3}$ and $x \leq \frac{1}{6}$ .

$- \frac{1}{3} < x \leq \frac{1}{6}$

Step 3

Solve $6 x - 12 < - 8$ when $x > \frac{1}{6}$ .

Solve $6 x - 12 < - 8$ for $x$ .

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

Add $12$ to both sides of the inequality.

$6 x < - 8 + 12$

Add $- 8$ and $12$ .

$6 x < 4$

Divide each term in $6 x < 4$ by $6$ and simplify.

Divide each term in $6 x < 4$ by $6$ .

$\frac{6 x}{6} < \frac{4}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} < \frac{4}{6}$

Divide $x$ by $1$ .

$x < \frac{4}{6}$

Simplify the right side.

Cancel the common factor of $4$ and $6$ .

Factor $2$ out of $4$ .

$x < \frac{2 (2)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x < \frac{2 \cdot 2}{2 \cdot 3}$

Cancel the common factor.

$x < \frac{2 \cdot 2}{2 \cdot 3}$

Rewrite the expression.

$x < \frac{2}{3}$

Find the intersection of $x < \frac{2}{3}$ and $x > \frac{1}{6}$ .

$\frac{1}{6} < x < \frac{2}{3}$

Step 4

Find the union of the solutions.

$- \frac{1}{3} < x < \frac{2}{3}$

Step 5

Convert the inequality to interval notation.

$(- \frac{1}{3}, \frac{2}{3})$

Step 6

Convert to Interval Notation |1-6x|-11<-8

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