Algebra Examples

$| h + 3 | < 5$

Step 1

Write $| h + 3 | < 5$ as a piecewise.

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Step 1.1

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

$h + 3 \geq 0$

Step 1.2

Subtract $3$ from both sides of the inequality.

$h \geq - 3$

Step 1.3

In the piece where $h + 3$ is non-negative, remove the absolute value.

$h + 3 < 5$

Step 1.4

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

$h + 3 < 0$

Step 1.5

Subtract $3$ from both sides of the inequality.

$h < - 3$

Step 1.6

In the piece where $h + 3$ is negative, remove the absolute value and multiply by $- 1$ .

$- (h + 3) < 5$

Step 1.7

Write as a piecewise.

${\begin{cases} h + 3 < 5 & h \geq - 3 \\ - (h + 3) < 5 & h < - 3 \end{cases}$

Step 1.8

Simplify $- (h + 3) < 5$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} h + 3 < 5 & h \geq - 3 \\ - h - 1 \cdot 3 < 5 & h < - 3 \end{cases}$

Step 1.8.2

Multiply $- 1$ by $3$ .

${\begin{cases} h + 3 < 5 & h \geq - 3 \\ - h - 3 < 5 & h < - 3 \end{cases}$

Step 2

Solve $h + 3 < 5$ when $h \geq - 3$ .

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Step 2.1

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

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Step 2.1.1

Subtract $3$ from both sides of the inequality.

$h < 5 - 3$

Step 2.1.2

Subtract $3$ from $5$ .

$h < 2$

Step 2.2

Find the intersection of $h < 2$ and $h \geq - 3$ .

$- 3 \leq h < 2$

Step 3

Solve $- h - 3 < 5$ when $h < - 3$ .

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Step 3.1

Solve $- h - 3 < 5$ for $h$ .

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Step 3.1.1

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

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Step 3.1.1.1

Add $3$ to both sides of the inequality.

$- h < 5 + 3$

Step 3.1.1.2

Add $5$ and $3$ .

$- h < 8$

Step 3.1.2

Divide each term in $- h < 8$ by $- 1$ and simplify.

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Step 3.1.2.1

Divide each term in $- h < 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- h}{- 1} > \frac{8}{- 1}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{h}{1} > \frac{8}{- 1}$

Step 3.1.2.2.2

Divide $h$ by $1$ .

$h > \frac{8}{- 1}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $8$ by $- 1$ .

$h > - 8$

Step 3.2

Find the intersection of $h > - 8$ and $h < - 3$ .

$- 8 < h < - 3$

Step 4

Find the union of the solutions.

$- 8 < h < 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 8 < h < 2$

Interval Notation:

$(- 8, 2)$

Step 6