Solve the Rational Equation for x (x-4)/(x+5)<=0

$\frac{x - 4}{x + 5} \leq 0$

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

$x - 4 = 0$

$x + 5 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Subtract $5$ from both sides of the equation.

$x = - 5$

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

$x = 4$

$x = - 5$

Consolidate the solutions.

$x = 4, - 5$

Find the domain of $\frac{x - 4}{x + 5}$ .

Set the denominator in $\frac{x - 4}{x + 5}$ equal to $0$ to find where the expression is undefined.

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

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

$(- \infty, - 5) \cup (- 5, \infty)$

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 4$

$x > 4$

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.

$x = - 8$

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

$\frac{(- 8) - 4}{(- 8) + 5} \leq 0$

The left side $4$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 5 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $- 5 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{(0) - 4}{(0) + 5} \leq 0$

The left side $- 0.8$ is less than the right side $0$ , which means that the given statement is always true.

True

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.

$\frac{(6) - 4}{(6) + 5} \leq 0$

The left side $0.\overline{18}$ is greater 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

$- 5 < x < 4$ True

$x > 4$ False

$x < - 5$ False

$- 5 < x < 4$ True

$x > 4$ False

The solution consists of all of the true intervals.

$- 5 < x \leq 4$

The result can be shown in multiple forms.

Inequality Form:

$- 5 < x \leq 4$

Interval Notation:

$(- 5, 4]$

Solve the Rational Equation for x (x-4)/(x+5)<=0

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