Solve the Inequality for x (x+5)/(x+4)<=0

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

Step 1

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

$x + 5 = 0$

$x + 4 = 0$

Step 2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 3

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 4

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

$x = - 5$

$x = - 4$

Step 5

Consolidate the solutions.

$x = - 5, - 4$

Step 6

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

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

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

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

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

Step 7

Use each root to create test intervals.

$x < - 5$

$- 5 < x < - 4$

$x > - 4$

Step 8

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.

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

The left side $0.75$ 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.

$x = - 4.5$

Replace $x$ with $- 4.5$ in the original inequality.

$\frac{(- 4.5) + 5}{(- 4.5) + 4} \leq 0$

The left side $- 1$ 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.

$x = 0$

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

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

The left side $1.25$ 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

Step 9

The solution consists of all of the true intervals.

$- 5 \leq x < - 4$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$- 5 \leq x < - 4$

Interval Notation:

$[- 5, - 4)$

Step 11

Solve the Inequality for x (x+5)/(x+4)<=0

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