Solve for x |x-1|>=6

$| x - 1 | \geq 6$

Write $| x - 1 | \geq 6$ 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.

$x - 1 \geq 6$

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 $- 1$ .

$- (x - 1) \geq 6$

Write as a piecewise.

${\begin{cases} x - 1 \geq 6 & x \geq 1 \\ - (x - 1) \geq 6 & x < 1 \end{cases}$

Simplify $- (x - 1) \geq 6$ .

Apply the distributive property.

${\begin{cases} x - 1 \geq 6 & x \geq 1 \\ - x - - 1 \geq 6 & x < 1 \end{cases}$

Multiply $- 1$ by $- 1$ .

${\begin{cases} x - 1 \geq 6 & x \geq 1 \\ - x + 1 \geq 6 & x < 1 \end{cases}$

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

Add $1$ to both sides of the inequality.

$x \geq 6 + 1$

Add $6$ and $1$ .

$x \geq 7$

Solve $- x + 1 \geq 6$ for $x$ .

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

Subtract $1$ from both sides of the inequality.

$- x \geq 6 - 1$

Subtract $1$ from $6$ .

$- x \geq 5$

Divide each term in $- x \geq 5$ by $- 1$ and simplify.

Divide each term in $- x \geq 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{5}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{5}{- 1}$

Simplify the right side.

Divide $5$ by $- 1$ .

$x \leq - 5$

Find the union of the solutions.

$x \leq - 5$ or $x \geq 7$

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 5 or x \geq 7$

Interval Notation:

$(- \infty, - 5] \cup [7, \infty)$

Solve for x |x-1|>=6

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