Solve the Absolute Value Inequality for x |x-1|>=2

$| x - 1 | \geq 2$

Write $| x - 1 | \geq 2$ 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.

$x - 1 \geq 2$

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 $- 1$ .

$- (x - 1) \geq 2$

Write as a piecewise.

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

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

Apply the distributive property.

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

Multiply $- 1$ by $- 1$ .

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

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

Add $1$ to both sides of the inequality.

$x \geq 2 + 1$

Add $2$ and $1$ .

$x \geq 3$

Solve $- x + 1 \geq 2$ for $x$ .

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

Subtract $1$ from both sides of the inequality.

$- x \geq 2 - 1$

Subtract $1$ from $2$ .

$- x \geq 1$

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

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

$\frac{- x}{- 1} \leq \frac{1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{1}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{1}{- 1}$

Simplify the right side.

Divide $1$ by $- 1$ .

$x \leq - 1$

Find the union of the solutions.

$x \leq - 1$ or $x \geq 3$

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 1 or x \geq 3$

Interval Notation:

$(- \infty, - 1] \cup [3, \infty)$

Solve the Absolute Value Inequality for x |x-1|>=2

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