Solve the Inequality for x (x-2)/(x-1)>=0

$\frac{x - 2}{x - 1} \geq 0$

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x - 2 = 0$

$x - 1 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Add $1$ to both sides of the equation.

$x = 1$

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 2$

$x = 1$

Consolidate the solutions.

$x = 2, 1$

Find the domain of $\frac{x - 2}{x - 1}$ .

Set the denominator in $\frac{x - 2}{x - 1}$ equal to $0$ to find where the expression is undefined.

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The domain is all values of $x$ that make the expression defined.

$(- \infty, 1) \cup (1, \infty)$

Use each root to create test intervals.

$x < 1$

$1 < x < 2$

$x > 2$

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < 1$ to see if it makes the inequality true.

Choose a value on the interval $x < 1$ and see if this value makes the original inequality true.

$x = 0$

Replace $x$ with $0$ in the original inequality.

$\frac{(0) - 2}{(0) - 1} \geq 0$

The left side $2$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $1 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $1 < x < 2$ and see if this value makes the original inequality true.

$x = 1.5$

Replace $x$ with $1.5$ in the original inequality.

$\frac{(1.5) - 2}{(1.5) - 1} \geq 0$

The left side $- 1$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 2$ to see if it makes the inequality true.

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

$\frac{(4) - 2}{(4) - 1} \geq 0$

The left side $0.\overline{6}$ is greater than the right side $0$ , which means that the given statement is always true.

True

Compare the intervals to determine which ones satisfy the original inequality.

$x < 1$ True

$1 < x < 2$ False

$x > 2$ True

$x < 1$ True

$1 < x < 2$ False

$x > 2$ True

The solution consists of all of the true intervals.

$x < 1$ or $x \geq 2$

The result can be shown in multiple forms.

Inequality Form:

$x < 1 or x \geq 2$

Interval Notation:

$(- \infty, 1) \cup [2, \infty)$

Solve the Inequality for x (x-2)/(x-1)>=0

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