Solve the Inequality for x (|x-4|)/5<=2

$\frac{| x - 4 |}{5} \leq 2$

Step 1

Multiply both sides by $5$ .

$\frac{| x - 4 |}{5} \cdot 5 \leq 2 \cdot 5$

Step 2

Simplify.

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{| x - 4 |}{5} \cdot 5 \leq 2 \cdot 5$

Rewrite the expression.

$| x - 4 | \leq 2 \cdot 5$

Simplify the right side.

Multiply $2$ by $5$ .

$| x - 4 | \leq 10$

Step 3

Solve for $x$ .

Write $| x - 4 | \leq 10$ 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.

$x - 4 \leq 10$

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 $- 1$ .

$- (x - 4) \leq 10$

Write as a piecewise.

${\begin{cases} x - 4 \leq 10 & x \geq 4 \\ - (x - 4) \leq 10 & x < 4 \end{cases}$

Simplify $- (x - 4) \leq 10$ .

Apply the distributive property.

${\begin{cases} x - 4 \leq 10 & x \geq 4 \\ - x - - 4 \leq 10 & x < 4 \end{cases}$

Multiply $- 1$ by $- 4$ .

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

Solve $x - 4 \leq 10$ when $x \geq 4$ .

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

Add $4$ to both sides of the inequality.

$x \leq 10 + 4$

Add $10$ and $4$ .

$x \leq 14$

Find the intersection of $x \leq 14$ and $x \geq 4$ .

$4 \leq x \leq 14$

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

Solve $- x + 4 \leq 10$ for $x$ .

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

Subtract $4$ from both sides of the inequality.

$- x \leq 10 - 4$

Subtract $4$ from $10$ .

$- x \leq 6$

Divide each term in $- x \leq 6$ by $- 1$ and simplify.

Divide each term in $- x \leq 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{6}{- 1}$

Divide $x$ by $1$ .

$x \geq \frac{6}{- 1}$

Simplify the right side.

Divide $6$ by $- 1$ .

$x \geq - 6$

Find the intersection of $x \geq - 6$ and $x < 4$ .

$- 6 \leq x < 4$

Find the union of the solutions.

$- 6 \leq x \leq 14$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- 6 \leq x \leq 14$

Interval Notation:

$[- 6, 14]$

Step 5

Solve the Inequality for x (|x-4|)/5<=2

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