Solve for x |x+2|+5<4

$| x + 2 | + 5 < 4$

Step 1

Write $| x + 2 | + 5 < 4$ as a piecewise.

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

$x + 2 \geq 0$

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

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

$x + 2 + 5 < 4$

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

$x + 2 < 0$

Subtract $2$ from both sides of the inequality.

$x < - 2$

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

$- (x + 2) + 5 < 4$

Write as a piecewise.

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

Add $2$ and $5$ .

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

Simplify $- (x + 2) + 5 < 4$ .

Simplify each term.

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

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

Add $- 2$ and $5$ .

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

Step 2

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

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

Subtract $7$ from both sides of the inequality.

$x < 4 - 7$

Subtract $7$ from $4$ .

$x < - 3$

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

No solution

Step 3

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

Solve $- x + 3 < 4$ for $x$ .

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

Subtract $3$ from both sides of the inequality.

$- x < 4 - 3$

Subtract $3$ from $4$ .

$- x < 1$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $1$ by $- 1$ .

$x > - 1$

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

No solution

Step 4

Find the union of the solutions.

No solution

Solve for x |x+2|+5<4

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