Solve the Inequality for x (-x+7)/(x-9)>=0

$\frac{- x + 7}{x - 9} \geq 0$

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

$- x + 7 = 0$

$x - 9 = 0$

Subtract $7$ from both sides of the equation.

$- x = - 7$

Divide each term in $- x = - 7$ by $- 1$ and simplify.

Divide each term in $- x = - 7$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 7}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 7}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 7}{- 1}$

Simplify the right side.

Divide $- 7$ by $- 1$ .

$x = 7$

Add $9$ to both sides of the equation.

$x = 9$

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

$x = 7$

$x = 9$

Consolidate the solutions.

$x = 7, 9$

Find the domain of $\frac{- x + 7}{x - 9}$ .

Set the denominator in $\frac{- x + 7}{x - 9}$ equal to $0$ to find where the expression is undefined.

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

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

$(- \infty, 9) \cup (9, \infty)$

Use each root to create test intervals.

$x < 7$

$7 < x < 9$

$x > 9$

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 $x < 7$ to see if it makes the inequality true.

Choose a value on the interval $x < 7$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{- (0) + 7}{(0) - 9} \geq 0$

The left side $- 0.\overline{7}$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $7 < x < 9$ to see if it makes the inequality true.

Choose a value on the interval $7 < x < 9$ and see if this value makes the original inequality true.

$x = 8$

Replace $x$ with $8$ in the original inequality.

$\frac{- (8) + 7}{(8) - 9} \geq 0$

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 $x > 9$ to see if it makes the inequality true.

Choose a value on the interval $x > 9$ and see if this value makes the original inequality true.

$x = 12$

Replace $x$ with $12$ in the original inequality.

$\frac{- (12) + 7}{(12) - 9} \geq 0$

The left side $- 1.\overline{6}$ 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.

$x < 7$ False

$7 < x < 9$ True

$x > 9$ False

$x < 7$ False

$7 < x < 9$ True

$x > 9$ False

The solution consists of all of the true intervals.

$7 \leq x < 9$

The result can be shown in multiple forms.

Inequality Form:

$7 \leq x < 9$

Interval Notation:

$[7, 9)$

Solve the Inequality for x (-x+7)/(x-9)>=0

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