Algebra Examples

$| x + 2 | + 5 \leq 4$

Step 1

Write $| x + 2 | + 5 \leq 4$ 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.

$x + 2 \geq 0$

Step 1.2

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

Step 1.3

In the piece where $x + 2$ is non-negative, remove the absolute value.

$x + 2 + 5 \leq 4$

Step 1.4

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

$x + 2 < 0$

Step 1.5

Subtract $2$ from both sides of the inequality.

$x < - 2$

Step 1.6

In the piece where $x + 2$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x + 2) + 5 \leq 4$

Step 1.7

Write as a piecewise.

${\begin{cases} x + 2 + 5 \leq 4 & x \geq - 2 \\ - (x + 2) + 5 \leq 4 & x < - 2 \end{cases}$

Step 1.8

Add $2$ and $5$ .

${\begin{cases} x + 7 \leq 4 & x \geq - 2 \\ - (x + 2) + 5 \leq 4 & x < - 2 \end{cases}$

Step 1.9

Simplify $- (x + 2) + 5 \leq 4$ .

Tap for more steps...

Step 1.9.1

Simplify each term.

Tap for more steps...

Step 1.9.1.1

Apply the distributive property.

${\begin{cases} x + 7 \leq 4 & x \geq - 2 \\ - x - 1 \cdot 2 + 5 \leq 4 & x < - 2 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $2$ .

${\begin{cases} x + 7 \leq 4 & x \geq - 2 \\ - x - 2 + 5 \leq 4 & x < - 2 \end{cases}$

Step 1.9.2

Add $- 2$ and $5$ .

${\begin{cases} x + 7 \leq 4 & x \geq - 2 \\ - x + 3 \leq 4 & x < - 2 \end{cases}$

Step 2

Solve $x + 7 \leq 4$ when $x \geq - 2$ .

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

Subtract $7$ from both sides of the inequality.

$x \leq 4 - 7$

Step 2.1.2

Subtract $7$ from $4$ .

$x \leq - 3$

Step 2.2

Find the intersection of $x \leq - 3$ and $x \geq - 2$ .

No solution

Step 3

Solve $- x + 3 \leq 4$ when $x < - 2$ .

Tap for more steps...

Step 3.1

Solve $- x + 3 \leq 4$ for $x$ .

Tap for more steps...

Step 3.1.1

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

Tap for more steps...

Step 3.1.1.1

Subtract $3$ from both sides of the inequality.

$- x \leq 4 - 3$

Step 3.1.1.2

Subtract $3$ from $4$ .

$- x \leq 1$

Step 3.1.2

Divide each term in $- x \leq 1$ by $- 1$ and simplify.

Tap for more steps...

Step 3.1.2.1

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

$\frac{- x}{- 1} \geq \frac{1}{- 1}$

Step 3.1.2.2

Simplify the left side.

Tap for more steps...

Step 3.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{1}{- 1}$

Step 3.1.2.2.2

Divide $x$ by $1$ .

$x \geq \frac{1}{- 1}$

Step 3.1.2.3

Simplify the right side.

Tap for more steps...

Step 3.1.2.3.1

Divide $1$ by $- 1$ .

$x \geq - 1$

Step 3.2

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

No solution

Step 4

Find the union of the solutions.

No solution