Solve the Inequality for x (x-2)/(x-4)>0

$\frac{x - 2}{x - 4} > 0$

Step 1

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

$x - 2 = 0$

$x - 4 = 0$

Step 2

Add $2$ to both sides of the equation.

$x = 2$

Step 3

Add $4$ to both sides of the equation.

$x = 4$

Step 4

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

$x = 2$

$x = 4$

Step 5

Consolidate the solutions.

$x = 2, 4$

Step 6

Find the domain of $\frac{x - 2}{x - 4}$ .

Set the denominator in $\frac{x - 2}{x - 4}$ equal to $0$ to find where the expression is undefined.

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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

$(- \infty, 4) \cup (4, \infty)$

Step 7

Use each root to create test intervals.

$x < 2$

$2 < x < 4$

$x > 4$

Step 8

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 $x < 2$ 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.

$x = 0$

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

$\frac{(0) - 2}{(0) - 4} > 0$

The left side $0.5$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $2 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $2 < x < 4$ and see if this value makes the original inequality true.

$x = 3$

Replace $x$ with $3$ in the original inequality.

$\frac{(3) - 2}{(3) - 4} > 0$

The left side $- 1$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 4$ to see if it makes the inequality true.

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Replace $x$ with $6$ in the original inequality.

$\frac{(6) - 2}{(6) - 4} > 0$

The left side $2$ 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.

$x < 2$ True

$2 < x < 4$ False

$x > 4$ True

$x < 2$ True

$2 < x < 4$ False

$x > 4$ True

Step 9

The solution consists of all of the true intervals.

$x < 2$ or $x > 4$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x < 2 or x > 4$

Interval Notation:

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

Step 11

Solve the Inequality for x (x-2)/(x-4)>0

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