Solve the Inequality for x |x-10|>=1

$| x - 10 | \geq 1$

Step 1

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

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

$x - 10 \geq 0$

Add $10$ to both sides of the inequality.

$x \geq 10$

In the piece where $x - 10$ is non-negative, remove the absolute value.

$x - 10 \geq 1$

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

$x - 10 < 0$

Add $10$ to both sides of the inequality.

$x < 10$

In the piece where $x - 10$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - 10) \geq 1$

Write as a piecewise.

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

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

Apply the distributive property.

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

Multiply $- 1$ by $- 10$ .

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

Step 2

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

Add $10$ to both sides of the inequality.

$x \geq 1 + 10$

Add $1$ and $10$ .

$x \geq 11$

Step 3

Solve $- x + 10 \geq 1$ for $x$ .

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

Subtract $10$ from both sides of the inequality.

$- x \geq 1 - 10$

Subtract $10$ from $1$ .

$- x \geq - 9$

Divide each term in $- x \geq - 9$ by $- 1$ and simplify.

Divide each term in $- x \geq - 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 9}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 9}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{- 9}{- 1}$

Simplify the right side.

Divide $- 9$ by $- 1$ .

$x \leq 9$

Step 4

Find the union of the solutions.

$x \leq 9$ or $x \geq 11$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq 9 or x \geq 11$

Interval Notation:

$(- \infty, 9] \cup [11, \infty)$

Step 6

Solve the Inequality for x |x-10|>=1

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