Graph |x+3|-2<=0

$| x + 3 | - 2 \leq 0$

Step 1

Write $| x + 3 | - 2 \leq 0$ as a piecewise.

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

$x + 3 \geq 0$

Subtract $3$ from both sides of the inequality.

$x \geq - 3$

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

$x + 3 - 2 \leq 0$

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

$x + 3 < 0$

Subtract $3$ from both sides of the inequality.

$x < - 3$

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

$- (x + 3) - 2 \leq 0$

Write as a piecewise.

${\begin{cases} x + 3 - 2 \leq 0 & x \geq - 3 \\ - (x + 3) - 2 \leq 0 & x < - 3 \end{cases}$

Subtract $2$ from $3$ .

${\begin{cases} x + 1 \leq 0 & x \geq - 3 \\ - (x + 3) - 2 \leq 0 & x < - 3 \end{cases}$

Simplify $- (x + 3) - 2 \leq 0$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} x + 1 \leq 0 & x \geq - 3 \\ - x - 1 \cdot 3 - 2 \leq 0 & x < - 3 \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} x + 1 \leq 0 & x \geq - 3 \\ - x - 3 - 2 \leq 0 & x < - 3 \end{cases}$

Subtract $2$ from $- 3$ .

${\begin{cases} x + 1 \leq 0 & x \geq - 3 \\ - x - 5 \leq 0 & x < - 3 \end{cases}$

Step 2

Solve $x + 1 \leq 0$ when $x \geq - 3$ .

Subtract $1$ from both sides of the inequality.

$x \leq - 1$

Find the intersection of $x \leq - 1$ and $x \geq - 3$ .

$- 3 \leq x \leq - 1$

Step 3

Solve $- x - 5 \leq 0$ when $x < - 3$ .

Solve $- x - 5 \leq 0$ for $x$ .

Add $5$ to both sides of the inequality.

$- x \leq 5$

Divide each term in $- x \leq 5$ by $- 1$ and simplify.

Divide each term in $- x \leq 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{5}{- 1}$

Divide $x$ by $1$ .

$x \geq \frac{5}{- 1}$

Simplify the right side.

Divide $5$ by $- 1$ .

$x \geq - 5$

Find the intersection of $x \geq - 5$ and $x < - 3$ .

$- 5 \leq x < - 3$

Step 4

Find the union of the solutions.

$- 5 \leq x \leq - 1$

Step 5

Graph |x+3|-2<=0

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