Solve the Inequality for x |11-x|<20

$| 11 - x | < 20$

Step 1

Write $| 11 - x | < 20$ as a piecewise.

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

$11 - x \geq 0$

Solve the inequality.

Subtract $11$ from both sides of the inequality.

$- x \geq - 11$

Divide each term in $- x \geq - 11$ by $- 1$ and simplify.

Divide each term in $- x \geq - 11$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 11}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 11}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{- 11}{- 1}$

Simplify the right side.

Divide $- 11$ by $- 1$ .

$x \leq 11$

In the piece where $11 - x$ is non-negative, remove the absolute value.

$11 - x < 20$

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

$11 - x < 0$

Solve the inequality.

Subtract $11$ from both sides of the inequality.

$- x < - 11$

Divide each term in $- x < - 11$ by $- 1$ and simplify.

Divide each term in $- x < - 11$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{- 11}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{- 11}{- 1}$

Divide $x$ by $1$ .

$x > \frac{- 11}{- 1}$

Simplify the right side.

Divide $- 11$ by $- 1$ .

$x > 11$

In the piece where $11 - x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (11 - x) < 20$

Write as a piecewise.

${\begin{cases} 11 - x < 20 & x \leq 11 \\ - (11 - x) < 20 & x > 11 \end{cases}$

Simplify $- (11 - x) < 20$ .

Apply the distributive property.

${\begin{cases} 11 - x < 20 & x \leq 11 \\ - 1 \cdot 11 - - x < 20 & x > 11 \end{cases}$

Multiply $- 1$ by $11$ .

${\begin{cases} 11 - x < 20 & x \leq 11 \\ - 11 - - x < 20 & x > 11 \end{cases}$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

${\begin{cases} 11 - x < 20 & x \leq 11 \\ - 11 + 1 x < 20 & x > 11 \end{cases}$

Multiply $x$ by $1$ .

${\begin{cases} 11 - x < 20 & x \leq 11 \\ - 11 + x < 20 & x > 11 \end{cases}$

Step 2

Solve $11 - x < 20$ when $x \leq 11$ .

Solve $11 - x < 20$ for $x$ .

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

Subtract $11$ from both sides of the inequality.

$- x < 20 - 11$

Subtract $11$ from $20$ .

$- x < 9$

Divide each term in $- x < 9$ by $- 1$ and simplify.

Divide each term in $- x < 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{9}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{9}{- 1}$

Divide $x$ by $1$ .

$x > \frac{9}{- 1}$

Simplify the right side.

Divide $9$ by $- 1$ .

$x > - 9$

Find the intersection of $x > - 9$ and $x \leq 11$ .

$- 9 < x \leq 11$

Step 3

Solve $- 11 + x < 20$ when $x > 11$ .

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

Add $11$ to both sides of the inequality.

$x < 20 + 11$

Add $20$ and $11$ .

$x < 31$

Find the intersection of $x < 31$ and $x > 11$ .

$11 < x < 31$

Step 4

Find the union of the solutions.

$- 9 < x < 31$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 9 < x < 31$

Interval Notation:

$(- 9, 31)$

Step 6

Solve the Inequality for x |11-x|<20

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