Solve the Inequality for x |x-8|<=5

$| x - 8 | \leq 5$

Step 1

Write $| x - 8 | \leq 5$ as a piecewise.

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

$x - 8 \geq 0$

Add $8$ to both sides of the inequality.

$x \geq 8$

In the piece where $x - 8$ is non-negative, remove the absolute value.

$x - 8 \leq 5$

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

$x - 8 < 0$

Add $8$ to both sides of the inequality.

$x < 8$

In the piece where $x - 8$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - 8) \leq 5$

Write as a piecewise.

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

Simplify $- (x - 8) \leq 5$ .

Apply the distributive property.

${\begin{cases} x - 8 \leq 5 & x \geq 8 \\ - x - - 8 \leq 5 & x < 8 \end{cases}$

Multiply $- 1$ by $- 8$ .

${\begin{cases} x - 8 \leq 5 & x \geq 8 \\ - x + 8 \leq 5 & x < 8 \end{cases}$

Step 2

Solve $x - 8 \leq 5$ when $x \geq 8$ .

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

Add $8$ to both sides of the inequality.

$x \leq 5 + 8$

Add $5$ and $8$ .

$x \leq 13$

Find the intersection of $x \leq 13$ and $x \geq 8$ .

$8 \leq x \leq 13$

Step 3

Solve $- x + 8 \leq 5$ when $x < 8$ .

Solve $- x + 8 \leq 5$ for $x$ .

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

Subtract $8$ from both sides of the inequality.

$- x \leq 5 - 8$

Subtract $8$ from $5$ .

$- x \leq - 3$

Divide each term in $- x \leq - 3$ by $- 1$ and simplify.

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 3$ by $- 1$ .

$x \geq 3$

Find the intersection of $x \geq 3$ and $x < 8$ .

$3 \leq x < 8$

Step 4

Find the union of the solutions.

$3 \leq x \leq 13$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$3 \leq x \leq 13$

Interval Notation:

$[3, 13]$

Step 6

Solve the Inequality for x |x-8|<=5

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