Solve the Absolute Value Inequality for b |b+3|>=7

$| b + 3 | \geq 7$

Step 1

Write $| b + 3 | \geq 7$ as a piecewise.

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

$b + 3 \geq 0$

Subtract $3$ from both sides of the inequality.

$b \geq - 3$

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

$b + 3 \geq 7$

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

$b + 3 < 0$

Subtract $3$ from both sides of the inequality.

$b < - 3$

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

$- (b + 3) \geq 7$

Write as a piecewise.

${\begin{cases} b + 3 \geq 7 & b \geq - 3 \\ - (b + 3) \geq 7 & b < - 3 \end{cases}$

Simplify $- (b + 3) \geq 7$ .

Apply the distributive property.

${\begin{cases} b + 3 \geq 7 & b \geq - 3 \\ - b - 1 \cdot 3 \geq 7 & b < - 3 \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} b + 3 \geq 7 & b \geq - 3 \\ - b - 3 \geq 7 & b < - 3 \end{cases}$

Step 2

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

Subtract $3$ from both sides of the inequality.

$b \geq 7 - 3$

Subtract $3$ from $7$ .

$b \geq 4$

Step 3

Solve $- b - 3 \geq 7$ for $b$ .

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

Add $3$ to both sides of the inequality.

$- b \geq 7 + 3$

Add $7$ and $3$ .

$- b \geq 10$

Divide each term in $- b \geq 10$ by $- 1$ and simplify.

Divide each term in $- b \geq 10$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- b}{- 1} \leq \frac{10}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{b}{1} \leq \frac{10}{- 1}$

Divide $b$ by $1$ .

$b \leq \frac{10}{- 1}$

Simplify the right side.

Divide $10$ by $- 1$ .

$b \leq - 10$

Step 4

Find the union of the solutions.

$b \leq - 10$ or $b \geq 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$b \leq - 10 or b \geq 4$

Interval Notation:

$(- \infty, - 10] \cup [4, \infty)$

Step 6

Solve the Absolute Value Inequality for b |b+3|>=7

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