Algebra Examples

$1 + | 1 + b | < 4$

Step 1

Write $1 + | 1 + b | < 4$ as a piecewise.

Tap for more steps...

Step 1.1

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

$1 + b \geq 0$

Step 1.2

Subtract $1$ from both sides of the inequality.

$b \geq - 1$

Step 1.3

In the piece where $1 + b$ is non-negative, remove the absolute value.

$1 + 1 + b < 4$

Step 1.4

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

$1 + b < 0$

Step 1.5

Subtract $1$ from both sides of the inequality.

$b < - 1$

Step 1.6

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

$1 - (1 + b) < 4$

Step 1.7

Write as a piecewise.

${\begin{cases} 1 + 1 + b < 4 & b \geq - 1 \\ 1 - (1 + b) < 4 & b < - 1 \end{cases}$

Step 1.8

Add $1$ and $1$ .

${\begin{cases} 2 + b < 4 & b \geq - 1 \\ 1 - (1 + b) < 4 & b < - 1 \end{cases}$

Step 1.9

Simplify $1 - (1 + b) < 4$ .

Tap for more steps...

Step 1.9.1

Simplify each term.

Tap for more steps...

Step 1.9.1.1

Apply the distributive property.

${\begin{cases} 2 + b < 4 & b \geq - 1 \\ 1 - 1 \cdot 1 - b < 4 & b < - 1 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $1$ .

${\begin{cases} 2 + b < 4 & b \geq - 1 \\ 1 - 1 - b < 4 & b < - 1 \end{cases}$

Step 1.9.2

Subtract $1$ from $1$ .

${\begin{cases} 2 + b < 4 & b \geq - 1 \\ 0 - b < 4 & b < - 1 \end{cases}$

Step 1.9.3

Subtract $b$ from $0$ .

${\begin{cases} 2 + b < 4 & b \geq - 1 \\ - b < 4 & b < - 1 \end{cases}$

Step 2

Solve $2 + b < 4$ when $b \geq - 1$ .

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

Subtract $2$ from both sides of the inequality.

$b < 4 - 2$

Step 2.1.2

Subtract $2$ from $4$ .

$b < 2$

Step 2.2

Find the intersection of $b < 2$ and $b \geq - 1$ .

$- 1 \leq b < 2$

Step 3

Solve $- b < 4$ when $b < - 1$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

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 3.1.2

Simplify the left side.

Tap for more steps...

Step 3.1.2.1

Dividing two negative values results in a positive value.

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

Step 3.1.2.2

Divide $b$ by $1$ .

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

Step 3.1.3

Simplify the right side.

Tap for more steps...

Step 3.1.3.1

Divide $4$ by $- 1$ .

$b > - 4$

Step 3.2

Find the intersection of $b > - 4$ and $b < - 1$ .

$- 4 < b < - 1$

Step 4

Find the union of the solutions.

$- 4 < b < 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 4 < b < 2$

Interval Notation:

$(- 4, 2)$

Step 6