Solve the Inequality for x (2|x-1|)/5>=2

$\frac{2 | x - 1 |}{5} \geq 2$

Multiply both sides by $5$ .

$\frac{2 | x - 1 |}{5} \cdot 5 \geq 2 \cdot 5$

Simplify.

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{2 | x - 1 |}{5} \cdot 5 \geq 2 \cdot 5$

Rewrite the expression.

$2 | x - 1 | \geq 2 \cdot 5$

Simplify the right side.

Multiply $2$ by $5$ .

$2 | x - 1 | \geq 10$

Solve for $x$ .

Write $2 | x - 1 | \geq 10$ as a piecewise.

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

$x - 1 \geq 0$

Add $1$ to both sides of the inequality.

$x \geq 1$

In the piece where $x - 1$ is non-negative, remove the absolute value.

$2 (x - 1) \geq 10$

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

$x - 1 < 0$

Add $1$ to both sides of the inequality.

$x < 1$

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

$2 (- (x - 1)) \geq 10$

Write as a piecewise.

${\begin{cases} 2 (x - 1) \geq 10 & x \geq 1 \\ 2 (- (x - 1)) \geq 10 & x < 1 \end{cases}$

Simplify $2 (x - 1) \geq 10$ .

Apply the distributive property.

${\begin{cases} 2 x + 2 \cdot - 1 \geq 10 & x \geq 1 \\ 2 (- (x - 1)) \geq 10 & x < 1 \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x - 2 \geq 10 & x \geq 1 \\ 2 (- (x - 1)) \geq 10 & x < 1 \end{cases}$

Simplify $2 (- (x - 1)) \geq 10$ .

Apply the distributive property.

${\begin{cases} 2 x - 2 \geq 10 & x \geq 1 \\ 2 (- x - - 1) \geq 10 & x < 1 \end{cases}$

Multiply $- 1$ by $- 1$ .

${\begin{cases} 2 x - 2 \geq 10 & x \geq 1 \\ 2 (- x + 1) \geq 10 & x < 1 \end{cases}$

Apply the distributive property.

${\begin{cases} 2 x - 2 \geq 10 & x \geq 1 \\ 2 (- x) + 2 \cdot 1 \geq 10 & x < 1 \end{cases}$

Multiply $- 1$ by $2$ .

${\begin{cases} 2 x - 2 \geq 10 & x \geq 1 \\ - 2 x + 2 \cdot 1 \geq 10 & x < 1 \end{cases}$

Multiply $2$ by $1$ .

${\begin{cases} 2 x - 2 \geq 10 & x \geq 1 \\ - 2 x + 2 \geq 10 & x < 1 \end{cases}$

Solve $2 x - 2 \geq 10$ for $x$ .

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

Add $2$ to both sides of the inequality.

$2 x \geq 10 + 2$

Add $10$ and $2$ .

$2 x \geq 12$

Divide each term in $2 x \geq 12$ by $2$ and simplify.

Divide each term in $2 x \geq 12$ by $2$ .

$\frac{2 x}{2} \geq \frac{12}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{12}{2}$

Divide $x$ by $1$ .

$x \geq \frac{12}{2}$

Simplify the right side.

Divide $12$ by $2$ .

$x \geq 6$

Solve $- 2 x + 2 \geq 10$ for $x$ .

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

Subtract $2$ from both sides of the inequality.

$- 2 x \geq 10 - 2$

Subtract $2$ from $10$ .

$- 2 x \geq 8$

Divide each term in $- 2 x \geq 8$ by $- 2$ and simplify.

Divide each term in $- 2 x \geq 8$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{8}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{8}{- 2}$

Divide $x$ by $1$ .

$x \leq \frac{8}{- 2}$

Simplify the right side.

Divide $8$ by $- 2$ .

$x \leq - 4$

Find the union of the solutions.

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

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

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

Solve the Inequality for x (2|x-1|)/5>=2

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