Solve the Inequality for x (-x+5)/(x-7)>=0

$\frac{- x + 5}{x - 7} \geq 0$

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

$- x + 5 = 0$

$x - 7 = 0$

Subtract $5$ from both sides of the equation.

$- x = - 5$

Divide each term in $- x = - 5$ by $- 1$ and simplify.

Divide each term in $- x = - 5$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 5}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 5}{- 1}$

Simplify the right side.

Divide $- 5$ by $- 1$ .

$x = 5$

Add $7$ to both sides of the equation.

$x = 7$

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

$x = 5$

$x = 7$

Consolidate the solutions.

$x = 5, 7$

Find the domain of $\frac{- x + 5}{x - 7}$ .

Set the denominator in $\frac{- x + 5}{x - 7}$ equal to $0$ to find where the expression is undefined.

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

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

$(- \infty, 7) \cup (7, \infty)$

Use each root to create test intervals.

$x < 5$

$5 < x < 7$

$x > 7$

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 $x < 5$ to see if it makes the inequality true.

Choose a value on the interval $x < 5$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{- (0) + 5}{(0) - 7} \geq 0$

The left side $- 0.\overline{714285}$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $5 < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $5 < x < 7$ and see if this value makes the original inequality true.

$x = 6$

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

$\frac{- (6) + 5}{(6) - 7} \geq 0$

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 $x > 7$ 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.

$x = 10$

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

$\frac{- (10) + 5}{(10) - 7} \geq 0$

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.

$x < 5$ False

$5 < x < 7$ True

$x > 7$ False

$x < 5$ False

$5 < x < 7$ True

$x > 7$ False

The solution consists of all of the true intervals.

$5 \leq x < 7$

The result can be shown in multiple forms.

Inequality Form:

$5 \leq x < 7$

Interval Notation:

$[5, 7)$

Solve the Inequality for x (-x+5)/(x-7)>=0

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