Solve the Inequality for x (2x-8)/((x-2)(x-8))<0

$\frac{2 x - 8}{(x - 2) (x - 8)} < 0$

Step 1

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

$2 x - 8 = 0$

$x - 2 = 0$

$x - 8 = 0$

Step 2

Add $8$ to both sides of the equation.

$2 x = 8$

Step 3

Divide each term in $2 x = 8$ by $2$ and simplify.

Divide each term in $2 x = 8$ by $2$ .

$\frac{2 x}{2} = \frac{8}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{8}{2}$

Divide $x$ by $1$ .

$x = \frac{8}{2}$

Simplify the right side.

Divide $8$ by $2$ .

$x = 4$

Step 4

Add $2$ to both sides of the equation.

$x = 2$

Step 5

Add $8$ to both sides of the equation.

$x = 8$

Step 6

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

$x = 4$

$x = 2$

$x = 8$

Step 7

Consolidate the solutions.

$x = 4, 2, 8$

Step 8

Find the domain of $\frac{2 x - 8}{(x - 2) (x - 8)}$ .

Set the denominator in $\frac{2 x - 8}{(x - 2) (x - 8)}$ equal to $0$ to find where the expression is undefined.

$(x - 2) (x - 8) = 0$

Solve for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x - 8 = 0$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

The final solution is all the values that make $(x - 2) (x - 8) = 0$ true.

$x = 2, 8$

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

$(- \infty, 2) \cup (2, 8) \cup (8, \infty)$

Step 9

Use each root to create test intervals.

$x < 2$

$2 < x < 4$

$4 < x < 8$

$x > 8$

Step 10

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.

$\frac{2 (0) - 8}{((0) - 2) ((0) - 8)} < 0$

The left side $- 0.5$ is less 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.

$\frac{2 (3) - 8}{((3) - 2) ((3) - 8)} < 0$

The left side $0.4$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $4 < x < 8$ to see if it makes the inequality true.

Choose a value on the interval $4 < x < 8$ and see if this value makes the original inequality true.

$x = 6$

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

$\frac{2 (6) - 8}{((6) - 2) ((6) - 8)} < 0$

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

True

Test a value on the interval $x > 8$ to see if it makes the inequality true.

Choose a value on the interval $x > 8$ and see if this value makes the original inequality true.

$x = 10$

Replace $x$ with $10$ in the original inequality.

$\frac{2 (10) - 8}{((10) - 2) ((10) - 8)} < 0$

The left side $0.75$ is not less than the right side $0$ , which means that the given statement is false.

False

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

$x < 2$ True

$2 < x < 4$ False

$4 < x < 8$ True

$x > 8$ False

$x < 2$ True

$2 < x < 4$ False

$4 < x < 8$ True

$x > 8$ False

Step 11

The solution consists of all of the true intervals.

$x < 2$ or $4 < x < 8$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$x < 2 or 4 < x < 8$

Interval Notation:

$(- \infty, 2) \cup (4, 8)$

Step 13

Solve the Inequality for x (2x-8)/((x-2)(x-8))<0

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