Solve for x 2|x-4|+2<=20

$2 | x - 4 | + 2 \leq 20$

Write $2 | x - 4 | + 2 \leq 20$ as a piecewise.

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

$x - 4 \geq 0$

Add $4$ to both sides of the inequality.

$x \geq 4$

In the piece where $x - 4$ is non-negative, remove the absolute value.

$2 (x - 4) + 2 \leq 20$

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

$x - 4 < 0$

Add $4$ to both sides of the inequality.

$x < 4$

In the piece where $x - 4$ is negative, remove the absolute value and multiply by $- 1$ .

$2 (- (x - 4)) + 2 \leq 20$

Write as a piecewise.

${\begin{cases} 2 (x - 4) + 2 \leq 20 & x \geq 4 \\ 2 (- (x - 4)) + 2 \leq 20 & x < 4 \end{cases}$

Simplify $2 (x - 4) + 2 \leq 20$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 2 x + 2 \cdot - 4 + 2 \leq 20 & x \geq 4 \\ 2 (- (x - 4)) + 2 \leq 20 & x < 4 \end{cases}$

Multiply $2$ by $- 4$ .

${\begin{cases} 2 x - 8 + 2 \leq 20 & x \geq 4 \\ 2 (- (x - 4)) + 2 \leq 20 & x < 4 \end{cases}$

Add $- 8$ and $2$ .

${\begin{cases} 2 x - 6 \leq 20 & x \geq 4 \\ 2 (- (x - 4)) + 2 \leq 20 & x < 4 \end{cases}$

Simplify $2 (- (x - 4)) + 2 \leq 20$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 2 x - 6 \leq 20 & x \geq 4 \\ 2 (- x - - 4) + 2 \leq 20 & x < 4 \end{cases}$

Multiply $- 1$ by $- 4$ .

${\begin{cases} 2 x - 6 \leq 20 & x \geq 4 \\ 2 (- x + 4) + 2 \leq 20 & x < 4 \end{cases}$

Apply the distributive property.

${\begin{cases} 2 x - 6 \leq 20 & x \geq 4 \\ 2 (- x) + 2 \cdot 4 + 2 \leq 20 & x < 4 \end{cases}$

Multiply $- 1$ by $2$ .

${\begin{cases} 2 x - 6 \leq 20 & x \geq 4 \\ - 2 x + 2 \cdot 4 + 2 \leq 20 & x < 4 \end{cases}$

Multiply $2$ by $4$ .

${\begin{cases} 2 x - 6 \leq 20 & x \geq 4 \\ - 2 x + 8 + 2 \leq 20 & x < 4 \end{cases}$

Add $8$ and $2$ .

${\begin{cases} 2 x - 6 \leq 20 & x \geq 4 \\ - 2 x + 10 \leq 20 & x < 4 \end{cases}$

Solve $2 x - 6 \leq 20$ when $x \geq 4$ .

Solve $2 x - 6 \leq 20$ for $x$ .

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

Add $6$ to both sides of the inequality.

$2 x \leq 20 + 6$

Add $20$ and $6$ .

$2 x \leq 26$

Divide each term in $2 x \leq 26$ by $2$ and simplify.

Divide each term in $2 x \leq 26$ by $2$ .

$\frac{2 x}{2} \leq \frac{26}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \leq \frac{26}{2}$

Divide $x$ by $1$ .

$x \leq \frac{26}{2}$

Simplify the right side.

Divide $26$ by $2$ .

$x \leq 13$

Find the intersection of $x \leq 13$ and $x \geq 4$ .

$4 \leq x \leq 13$

Solve $- 2 x + 10 \leq 20$ when $x < 4$ .

Solve $- 2 x + 10 \leq 20$ for $x$ .

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

Subtract $10$ from both sides of the inequality.

$- 2 x \leq 20 - 10$

Subtract $10$ from $20$ .

$- 2 x \leq 10$

Divide each term in $- 2 x \leq 10$ by $- 2$ and simplify.

Divide each term in $- 2 x \leq 10$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \geq \frac{10}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \geq \frac{10}{- 2}$

Divide $x$ by $1$ .

$x \geq \frac{10}{- 2}$

Simplify the right side.

Divide $10$ by $- 2$ .

$x \geq - 5$

Find the intersection of $x \geq - 5$ and $x < 4$ .

$- 5 \leq x < 4$

Find the union of the solutions.

$- 5 \leq x \leq 13$

The result can be shown in multiple forms.

Inequality Form:

$- 5 \leq x \leq 13$

Interval Notation:

$[- 5, 13]$

Solve for x 2|x-4|+2<=20

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