Solve for x |2x+2|-7<=-5

$| 2 x + 2 | - 7 \leq - 5$

Write $| 2 x + 2 | - 7 \leq - 5$ as a piecewise.

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

$2 x + 2 \geq 0$

Solve the inequality.

Subtract $2$ from both sides of the inequality.

$2 x \geq - 2$

Divide each term in $2 x \geq - 2$ by $2$ and simplify.

Divide each term in $2 x \geq - 2$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 2}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 2}{2}$

Divide $x$ by $1$ .

$x \geq \frac{- 2}{2}$

Simplify the right side.

Divide $- 2$ by $2$ .

$x \geq - 1$

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

$2 x + 2 - 7 \leq - 5$

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

$2 x + 2 < 0$

Solve the inequality.

Subtract $2$ from both sides of the inequality.

$2 x < - 2$

Divide each term in $2 x < - 2$ by $2$ and simplify.

Divide each term in $2 x < - 2$ by $2$ .

$\frac{2 x}{2} < \frac{- 2}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{- 2}{2}$

Divide $x$ by $1$ .

$x < \frac{- 2}{2}$

Simplify the right side.

Divide $- 2$ by $2$ .

$x < - 1$

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

$- (2 x + 2) - 7 \leq - 5$

Write as a piecewise.

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

Subtract $7$ from $2$ .

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

Simplify $- (2 x + 2) - 7 \leq - 5$ .

Simplify each term.

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $2$ .

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

Subtract $7$ from $- 2$ .

${\begin{cases} 2 x - 5 \leq - 5 & x \geq - 1 \\ - 2 x - 9 \leq - 5 & x < - 1 \end{cases}$

Solve $2 x - 5 \leq - 5$ when $x \geq - 1$ .

Solve $2 x - 5 \leq - 5$ for $x$ .

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

Add $5$ to both sides of the inequality.

$2 x \leq - 5 + 5$

Add $- 5$ and $5$ .

$2 x \leq 0$

Divide each term in $2 x \leq 0$ by $2$ and simplify.

Divide each term in $2 x \leq 0$ by $2$ .

$\frac{2 x}{2} \leq \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \leq \frac{0}{2}$

Divide $x$ by $1$ .

$x \leq \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x \leq 0$

Find the intersection of $x \leq 0$ and $x \geq - 1$ .

$- 1 \leq x \leq 0$

Solve $- 2 x - 9 \leq - 5$ when $x < - 1$ .

Solve $- 2 x - 9 \leq - 5$ for $x$ .

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

Add $9$ to both sides of the inequality.

$- 2 x \leq - 5 + 9$

Add $- 5$ and $9$ .

$- 2 x \leq 4$

Divide each term in $- 2 x \leq 4$ by $- 2$ and simplify.

Divide each term in $- 2 x \leq 4$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \geq \frac{4}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \geq \frac{4}{- 2}$

Divide $x$ by $1$ .

$x \geq \frac{4}{- 2}$

Simplify the right side.

Divide $4$ by $- 2$ .

$x \geq - 2$

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

$- 2 \leq x < - 1$

Find the union of the solutions.

$- 2 \leq x \leq 0$

The result can be shown in multiple forms.

Inequality Form:

$- 2 \leq x \leq 0$

Interval Notation:

$[- 2, 0]$

Solve for x |2x+2|-7<=-5

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