Solve for h |h-5|<=10

$| h - 5 | \leq 10$

Step 1

Write $| h - 5 | \leq 10$ as a piecewise.

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

$h - 5 \geq 0$

Add $5$ to both sides of the inequality.

$h \geq 5$

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

$h - 5 \leq 10$

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

$h - 5 < 0$

Add $5$ to both sides of the inequality.

$h < 5$

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

$- (h - 5) \leq 10$

Write as a piecewise.

${\begin{cases} h - 5 \leq 10 & h \geq 5 \\ - (h - 5) \leq 10 & h < 5 \end{cases}$

Simplify $- (h - 5) \leq 10$ .

Apply the distributive property.

${\begin{cases} h - 5 \leq 10 & h \geq 5 \\ - h - - 5 \leq 10 & h < 5 \end{cases}$

Multiply $- 1$ by $- 5$ .

${\begin{cases} h - 5 \leq 10 & h \geq 5 \\ - h + 5 \leq 10 & h < 5 \end{cases}$

Step 2

Solve $h - 5 \leq 10$ when $h \geq 5$ .

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

Add $5$ to both sides of the inequality.

$h \leq 10 + 5$

Add $10$ and $5$ .

$h \leq 15$

Find the intersection of $h \leq 15$ and $h \geq 5$ .

$5 \leq h \leq 15$

Step 3

Solve $- h + 5 \leq 10$ when $h < 5$ .

Solve $- h + 5 \leq 10$ for $h$ .

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

Subtract $5$ from both sides of the inequality.

$- h \leq 10 - 5$

Subtract $5$ from $10$ .

$- h \leq 5$

Divide each term in $- h \leq 5$ by $- 1$ and simplify.

Divide each term in $- h \leq 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- h}{- 1} \geq \frac{5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{h}{1} \geq \frac{5}{- 1}$

Divide $h$ by $1$ .

$h \geq \frac{5}{- 1}$

Simplify the right side.

Divide $5$ by $- 1$ .

$h \geq - 5$

Find the intersection of $h \geq - 5$ and $h < 5$ .

$- 5 \leq h < 5$

Step 4

Find the union of the solutions.

$- 5 \leq h \leq 15$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 5 \leq h \leq 15$

Interval Notation:

$[- 5, 15]$

Step 6

Solve for h |h-5|<=10

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