Solve for b 5|b+8|-7>13

$5 | b + 8 | - 7 > 13$

Step 1

Write $5 | b + 8 | - 7 > 13$ as a piecewise.

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

$b + 8 \geq 0$

Subtract $8$ from both sides of the inequality.

$b \geq - 8$

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

$5 (b + 8) - 7 > 13$

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

$b + 8 < 0$

Subtract $8$ from both sides of the inequality.

$b < - 8$

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

$5 (- (b + 8)) - 7 > 13$

Write as a piecewise.

${\begin{cases} 5 (b + 8) - 7 > 13 & b \geq - 8 \\ 5 (- (b + 8)) - 7 > 13 & b < - 8 \end{cases}$

Simplify $5 (b + 8) - 7 > 13$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 5 b + 5 \cdot 8 - 7 > 13 & b \geq - 8 \\ 5 (- (b + 8)) - 7 > 13 & b < - 8 \end{cases}$

Multiply $5$ by $8$ .

${\begin{cases} 5 b + 40 - 7 > 13 & b \geq - 8 \\ 5 (- (b + 8)) - 7 > 13 & b < - 8 \end{cases}$

Subtract $7$ from $40$ .

${\begin{cases} 5 b + 33 > 13 & b \geq - 8 \\ 5 (- (b + 8)) - 7 > 13 & b < - 8 \end{cases}$

Simplify $5 (- (b + 8)) - 7 > 13$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 5 b + 33 > 13 & b \geq - 8 \\ 5 (- b - 1 \cdot 8) - 7 > 13 & b < - 8 \end{cases}$

Multiply $- 1$ by $8$ .

${\begin{cases} 5 b + 33 > 13 & b \geq - 8 \\ 5 (- b - 8) - 7 > 13 & b < - 8 \end{cases}$

Apply the distributive property.

${\begin{cases} 5 b + 33 > 13 & b \geq - 8 \\ 5 (- b) + 5 \cdot - 8 - 7 > 13 & b < - 8 \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} 5 b + 33 > 13 & b \geq - 8 \\ - 5 b + 5 \cdot - 8 - 7 > 13 & b < - 8 \end{cases}$

Multiply $5$ by $- 8$ .

${\begin{cases} 5 b + 33 > 13 & b \geq - 8 \\ - 5 b - 40 - 7 > 13 & b < - 8 \end{cases}$

Subtract $7$ from $- 40$ .

${\begin{cases} 5 b + 33 > 13 & b \geq - 8 \\ - 5 b - 47 > 13 & b < - 8 \end{cases}$

Step 2

Solve $5 b + 33 > 13$ for $b$ .

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

Subtract $33$ from both sides of the inequality.

$5 b > 13 - 33$

Subtract $33$ from $13$ .

$5 b > - 20$

Divide each term in $5 b > - 20$ by $5$ and simplify.

Divide each term in $5 b > - 20$ by $5$ .

$\frac{5 b}{5} > \frac{- 20}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 b}{5} > \frac{- 20}{5}$

Divide $b$ by $1$ .

$b > \frac{- 20}{5}$

Simplify the right side.

Divide $- 20$ by $5$ .

$b > - 4$

Step 3

Solve $- 5 b - 47 > 13$ for $b$ .

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

Add $47$ to both sides of the inequality.

$- 5 b > 13 + 47$

Add $13$ and $47$ .

$- 5 b > 60$

Divide each term in $- 5 b > 60$ by $- 5$ and simplify.

Divide each term in $- 5 b > 60$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 b}{- 5} < \frac{60}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 b}{- 5} < \frac{60}{- 5}$

Divide $b$ by $1$ .

$b < \frac{60}{- 5}$

Simplify the right side.

Divide $60$ by $- 5$ .

$b < - 12$

Step 4

Find the union of the solutions.

$b < - 12$ or $b > - 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$b < - 12 or b > - 4$

Interval Notation:

$(- \infty, - 12) \cup (- 4, \infty)$

Step 6

Solve for b 5|b+8|-7>13

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