Finite Math Examples

$| b - 8 | + 10 > 22$

Step 1

Write $| b - 8 | + 10 > 22$ 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.

$b - 8 \geq 0$

Step 1.2

Add $8$ to both sides of the inequality.

$b \geq 8$

Step 1.3

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

$b - 8 + 10 > 22$

Step 1.4

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

$b - 8 < 0$

Step 1.5

Add $8$ to both sides of the inequality.

$b < 8$

Step 1.6

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

$- (b - 8) + 10 > 22$

Step 1.7

Write as a piecewise.

${\begin{cases} b - 8 + 10 > 22 & b \geq 8 \\ - (b - 8) + 10 > 22 & b < 8 \end{cases}$

Step 1.8

Add $- 8$ and $10$ .

${\begin{cases} b + 2 > 22 & b \geq 8 \\ - (b - 8) + 10 > 22 & b < 8 \end{cases}$

Step 1.9

Simplify $- (b - 8) + 10 > 22$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} b + 2 > 22 & b \geq 8 \\ - b - - 8 + 10 > 22 & b < 8 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $- 8$ .

${\begin{cases} b + 2 > 22 & b \geq 8 \\ - b + 8 + 10 > 22 & b < 8 \end{cases}$

Step 1.9.2

Add $8$ and $10$ .

${\begin{cases} b + 2 > 22 & b \geq 8 \\ - b + 18 > 22 & b < 8 \end{cases}$

Step 2

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

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Step 2.1

Subtract $2$ from both sides of the inequality.

$b > 22 - 2$

Step 2.2

Subtract $2$ from $22$ .

$b > 20$

Step 3

Solve $- b + 18 > 22$ for $b$ .

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Step 3.1

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

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Step 3.1.1

Subtract $18$ from both sides of the inequality.

$- b > 22 - 18$

Step 3.1.2

Subtract $18$ from $22$ .

$- b > 4$

Step 3.2

Divide each term in $- b > 4$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- b > 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- b}{- 1} < \frac{4}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{b}{1} < \frac{4}{- 1}$

Step 3.2.2.2

Divide $b$ by $1$ .

$b < \frac{4}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $4$ by $- 1$ .

$b < - 4$

Step 4

Find the union of the solutions.

$b < - 4$ or $b > 20$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$b < - 4 or b > 20$

Interval Notation:

$(- \infty, - 4) \cup (20, \infty)$

Step 6