Solve for x -5|2-2x|+8<=-42

$- 5 | 2 - 2 x | + 8 \leq - 42$

Step 1

Write $- 5 | 2 - 2 x | + 8 \leq - 42$ as a piecewise.

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

$2 - 2 x \geq 0$

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 $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 2$ by $- 2$ .

$x \leq 1$

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

$- 5 (2 - 2 x) + 8 \leq - 42$

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

$2 - 2 x < 0$

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 $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

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

$- 5 (- (2 - 2 x)) + 8 \leq - 42$

Write as a piecewise.

${\begin{cases} - 5 (2 - 2 x) + 8 \leq - 42 & x \leq 1 \\ - 5 (- (2 - 2 x)) + 8 \leq - 42 & x > 1 \end{cases}$

Simplify $- 5 (2 - 2 x) + 8 \leq - 42$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 5 \cdot 2 - 5 (- 2 x) + 8 \leq - 42 & x \leq 1 \\ - 5 (- (2 - 2 x)) + 8 \leq - 42 & x > 1 \end{cases}$

Multiply $- 5$ by $2$ .

${\begin{cases} - 10 - 5 (- 2 x) + 8 \leq - 42 & x \leq 1 \\ - 5 (- (2 - 2 x)) + 8 \leq - 42 & x > 1 \end{cases}$

Multiply $- 2$ by $- 5$ .

${\begin{cases} - 10 + 10 x + 8 \leq - 42 & x \leq 1 \\ - 5 (- (2 - 2 x)) + 8 \leq - 42 & x > 1 \end{cases}$

Add $- 10$ and $8$ .

${\begin{cases} 10 x - 2 \leq - 42 & x \leq 1 \\ - 5 (- (2 - 2 x)) + 8 \leq - 42 & x > 1 \end{cases}$

Simplify $- 5 (- (2 - 2 x)) + 8 \leq - 42$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 10 x - 2 \leq - 42 & x \leq 1 \\ - 5 (- 1 \cdot 2 - (- 2 x)) + 8 \leq - 42 & x > 1 \end{cases}$

Multiply $- 1$ by $2$ .

${\begin{cases} 10 x - 2 \leq - 42 & x \leq 1 \\ - 5 (- 2 - (- 2 x)) + 8 \leq - 42 & x > 1 \end{cases}$

Multiply $- 2$ by $- 1$ .

${\begin{cases} 10 x - 2 \leq - 42 & x \leq 1 \\ - 5 (- 2 + 2 x) + 8 \leq - 42 & x > 1 \end{cases}$

Apply the distributive property.

${\begin{cases} 10 x - 2 \leq - 42 & x \leq 1 \\ - 5 \cdot - 2 - 5 (2 x) + 8 \leq - 42 & x > 1 \end{cases}$

Multiply $- 5$ by $- 2$ .

${\begin{cases} 10 x - 2 \leq - 42 & x \leq 1 \\ 10 - 5 (2 x) + 8 \leq - 42 & x > 1 \end{cases}$

Multiply $2$ by $- 5$ .

${\begin{cases} 10 x - 2 \leq - 42 & x \leq 1 \\ 10 - 10 x + 8 \leq - 42 & x > 1 \end{cases}$

Add $10$ and $8$ .

${\begin{cases} 10 x - 2 \leq - 42 & x \leq 1 \\ - 10 x + 18 \leq - 42 & x > 1 \end{cases}$

Step 2

Solve $10 x - 2 \leq - 42$ for $x$ .

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

Add $2$ to both sides of the inequality.

$10 x \leq - 42 + 2$

Add $- 42$ and $2$ .

$10 x \leq - 40$

Divide each term in $10 x \leq - 40$ by $10$ and simplify.

Divide each term in $10 x \leq - 40$ by $10$ .

$\frac{10 x}{10} \leq \frac{- 40}{10}$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 x}{10} \leq \frac{- 40}{10}$

Divide $x$ by $1$ .

$x \leq \frac{- 40}{10}$

Simplify the right side.

Divide $- 40$ by $10$ .

$x \leq - 4$

Step 3

Solve $- 10 x + 18 \leq - 42$ for $x$ .

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

Subtract $18$ from both sides of the inequality.

$- 10 x \leq - 42 - 18$

Subtract $18$ from $- 42$ .

$- 10 x \leq - 60$

Divide each term in $- 10 x \leq - 60$ by $- 10$ and simplify.

Divide each term in $- 10 x \leq - 60$ by $- 10$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 10 x}{- 10} \geq \frac{- 60}{- 10}$

Simplify the left side.

Cancel the common factor of $- 10$ .

Cancel the common factor.

$\frac{- 10 x}{- 10} \geq \frac{- 60}{- 10}$

Divide $x$ by $1$ .

$x \geq \frac{- 60}{- 10}$

Simplify the right side.

Divide $- 60$ by $- 10$ .

$x \geq 6$

Step 4

Find the union of the solutions.

$x \leq - 4$ or $x \geq 6$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 4 or x \geq 6$

Interval Notation:

$(- \infty, - 4] \cup [6, \infty)$

Step 6

Solve for x -5|2-2x|+8<=-42

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