Solve the Inequality for b |b-7|<6

$| b - 7 | < 6$

Step 1

Write $| b - 7 | < 6$ as a piecewise.

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

$b - 7 \geq 0$

Add $7$ to both sides of the inequality.

$b \geq 7$

In the piece where $b - 7$ is non-negative, remove the absolute value.

$b - 7 < 6$

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

$b - 7 < 0$

Add $7$ to both sides of the inequality.

$b < 7$

In the piece where $b - 7$ is negative, remove the absolute value and multiply by $- 1$ .

$- (b - 7) < 6$

Write as a piecewise.

${\begin{cases} b - 7 < 6 & b \geq 7 \\ - (b - 7) < 6 & b < 7 \end{cases}$

Simplify $- (b - 7) < 6$ .

Apply the distributive property.

${\begin{cases} b - 7 < 6 & b \geq 7 \\ - b - - 7 < 6 & b < 7 \end{cases}$

Multiply $- 1$ by $- 7$ .

${\begin{cases} b - 7 < 6 & b \geq 7 \\ - b + 7 < 6 & b < 7 \end{cases}$

Step 2

Solve $b - 7 < 6$ when $b \geq 7$ .

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

Add $7$ to both sides of the inequality.

$b < 6 + 7$

Add $6$ and $7$ .

$b < 13$

Find the intersection of $b < 13$ and $b \geq 7$ .

$7 \leq b < 13$

Step 3

Solve $- b + 7 < 6$ when $b < 7$ .

Solve $- b + 7 < 6$ for $b$ .

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

Subtract $7$ from both sides of the inequality.

$- b < 6 - 7$

Subtract $7$ from $6$ .

$- b < - 1$

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

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

$\frac{- b}{- 1} > \frac{- 1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{b}{1} > \frac{- 1}{- 1}$

Divide $b$ by $1$ .

$b > \frac{- 1}{- 1}$

Simplify the right side.

Divide $- 1$ by $- 1$ .

$b > 1$

Find the intersection of $b > 1$ and $b < 7$ .

$1 < b < 7$

Step 4

Find the union of the solutions.

$1 < b < 13$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$1 < b < 13$

Interval Notation:

$(1, 13)$

Step 6

Solve the Inequality for b |b-7|<6

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