Solve the Inequality for x (-x+6)/(x-8)>=0

$\frac{- x + 6}{x - 8} \geq 0$

Step 1

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

$- x + 6 = 0$

$x - 8 = 0$

Step 2

Subtract $6$ from both sides of the equation.

$- x = - 6$

Step 3

Divide each term in $- x = - 6$ by $- 1$ and simplify.

Divide each term in $- x = - 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 6}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 6}{- 1}$

Simplify the right side.

Divide $- 6$ by $- 1$ .

$x = 6$

Step 4

Add $8$ to both sides of the equation.

$x = 8$

Step 5

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

$x = 6$

$x = 8$

Step 6

Consolidate the solutions.

$x = 6, 8$

Step 7

Find the domain of $\frac{- x + 6}{x - 8}$ .

Set the denominator in $\frac{- x + 6}{x - 8}$ equal to $0$ to find where the expression is undefined.

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

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

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

Step 8

Use each root to create test intervals.

$x < 6$

$6 < x < 8$

$x > 8$

Step 9

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 $x < 6$ to see if it makes the inequality true.

Choose a value on the interval $x < 6$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{- (0) + 6}{(0) - 8} \geq 0$

The left side $- 0.75$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $6 < x < 8$ to see if it makes the inequality true.

Choose a value on the interval $6 < x < 8$ and see if this value makes the original inequality true.

$x = 7$

Replace $x$ with $7$ in the original inequality.

$\frac{- (7) + 6}{(7) - 8} \geq 0$

The left side $1$ is greater 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.

$\frac{- (10) + 6}{(10) - 8} \geq 0$

The left side $- 2$ is 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.

$x < 6$ False

$6 < x < 8$ True

$x > 8$ False

$x < 6$ False

$6 < x < 8$ True

$x > 8$ False

Step 10

The solution consists of all of the true intervals.

$6 \leq x < 8$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$6 \leq x < 8$

Interval Notation:

$[6, 8)$

Step 12

Solve the Inequality for x (-x+6)/(x-8)>=0

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