Evaluate |x-1|>=3

$| x - 1 | \geq 3$

Step 1

Write $| x - 1 | \geq 3$ 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.

$x - 1 \geq 3$

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 $- 1$ .

$- (x - 1) \geq 3$

Write as a piecewise.

${\begin{cases} x - 1 \geq 3 & x \geq 1 \\ - (x - 1) \geq 3 & x < 1 \end{cases}$

Simplify $- (x - 1) \geq 3$ .

Apply the distributive property.

${\begin{cases} x - 1 \geq 3 & x \geq 1 \\ - x - - 1 \geq 3 & x < 1 \end{cases}$

Multiply $- 1$ by $- 1$ .

${\begin{cases} x - 1 \geq 3 & x \geq 1 \\ - x + 1 \geq 3 & x < 1 \end{cases}$

Step 2

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

Add $1$ to both sides of the inequality.

$x \geq 3 + 1$

Add $3$ and $1$ .

$x \geq 4$

Step 3

Solve $- x + 1 \geq 3$ for $x$ .

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

Subtract $1$ from both sides of the inequality.

$- x \geq 3 - 1$

Subtract $1$ from $3$ .

$- x \geq 2$

Divide each term in $- x \geq 2$ by $- 1$ and simplify.

Divide each term in $- x \geq 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{2}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{2}{- 1}$

Simplify the right side.

Divide $2$ by $- 1$ .

$x \leq - 2$

Step 4

Find the union of the solutions.

$x \leq - 2$ or $x \geq 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 2 or x \geq 4$

Interval Notation:

$(- \infty, - 2] \cup [4, \infty)$

Step 6

Evaluate |x-1|>=3

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