Solve the Absolute Value Inequality for x 5|x-5|+6<=11

$5 | x - 5 | + 6 \leq 11$

Step 1

Write $5 | x - 5 | + 6 \leq 11$ as a piecewise.

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

$x - 5 \geq 0$

Add $5$ to both sides of the inequality.

$x \geq 5$

In the piece where $x - 5$ is non-negative, remove the absolute value.

$5 (x - 5) + 6 \leq 11$

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

$x - 5 < 0$

Add $5$ to both sides of the inequality.

$x < 5$

In the piece where $x - 5$ is negative, remove the absolute value and multiply by $- 1$ .

$5 (- (x - 5)) + 6 \leq 11$

Write as a piecewise.

${\begin{cases} 5 (x - 5) + 6 \leq 11 & x \geq 5 \\ 5 (- (x - 5)) + 6 \leq 11 & x < 5 \end{cases}$

Simplify $5 (x - 5) + 6 \leq 11$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 5 x + 5 \cdot - 5 + 6 \leq 11 & x \geq 5 \\ 5 (- (x - 5)) + 6 \leq 11 & x < 5 \end{cases}$

Multiply $5$ by $- 5$ .

${\begin{cases} 5 x - 25 + 6 \leq 11 & x \geq 5 \\ 5 (- (x - 5)) + 6 \leq 11 & x < 5 \end{cases}$

Add $- 25$ and $6$ .

${\begin{cases} 5 x - 19 \leq 11 & x \geq 5 \\ 5 (- (x - 5)) + 6 \leq 11 & x < 5 \end{cases}$

Simplify $5 (- (x - 5)) + 6 \leq 11$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 5 x - 19 \leq 11 & x \geq 5 \\ 5 (- x - - 5) + 6 \leq 11 & x < 5 \end{cases}$

Multiply $- 1$ by $- 5$ .

${\begin{cases} 5 x - 19 \leq 11 & x \geq 5 \\ 5 (- x + 5) + 6 \leq 11 & x < 5 \end{cases}$

Apply the distributive property.

${\begin{cases} 5 x - 19 \leq 11 & x \geq 5 \\ 5 (- x) + 5 \cdot 5 + 6 \leq 11 & x < 5 \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} 5 x - 19 \leq 11 & x \geq 5 \\ - 5 x + 5 \cdot 5 + 6 \leq 11 & x < 5 \end{cases}$

Multiply $5$ by $5$ .

${\begin{cases} 5 x - 19 \leq 11 & x \geq 5 \\ - 5 x + 25 + 6 \leq 11 & x < 5 \end{cases}$

Add $25$ and $6$ .

${\begin{cases} 5 x - 19 \leq 11 & x \geq 5 \\ - 5 x + 31 \leq 11 & x < 5 \end{cases}$

Step 2

Solve $5 x - 19 \leq 11$ when $x \geq 5$ .

Solve $5 x - 19 \leq 11$ for $x$ .

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

Add $19$ to both sides of the inequality.

$5 x \leq 11 + 19$

Add $11$ and $19$ .

$5 x \leq 30$

Divide each term in $5 x \leq 30$ by $5$ and simplify.

Divide each term in $5 x \leq 30$ by $5$ .

$\frac{5 x}{5} \leq \frac{30}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} \leq \frac{30}{5}$

Divide $x$ by $1$ .

$x \leq \frac{30}{5}$

Simplify the right side.

Divide $30$ by $5$ .

$x \leq 6$

Find the intersection of $x \leq 6$ and $x \geq 5$ .

$5 \leq x \leq 6$

Step 3

Solve $- 5 x + 31 \leq 11$ when $x < 5$ .

Solve $- 5 x + 31 \leq 11$ for $x$ .

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

Subtract $31$ from both sides of the inequality.

$- 5 x \leq 11 - 31$

Subtract $31$ from $11$ .

$- 5 x \leq - 20$

Divide each term in $- 5 x \leq - 20$ by $- 5$ and simplify.

Divide each term in $- 5 x \leq - 20$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} \geq \frac{- 20}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} \geq \frac{- 20}{- 5}$

Divide $x$ by $1$ .

$x \geq \frac{- 20}{- 5}$

Simplify the right side.

Divide $- 20$ by $- 5$ .

$x \geq 4$

Find the intersection of $x \geq 4$ and $x < 5$ .

$4 \leq x < 5$

Step 4

Find the union of the solutions.

$4 \leq x \leq 6$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$4 \leq x \leq 6$

Interval Notation:

$[4, 6]$

Step 6

Solve the Absolute Value Inequality for x 5|x-5|+6<=11

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