Solve for x ((x+5)(x+1))/(x-4)>=0

$\frac{(x + 5) (x + 1)}{x - 4} \geq 0$

Step 1

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

$x + 5 = 0$

$x + 1 = 0$

$x - 4 = 0$

Step 2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 3

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4

Add $4$ to both sides of the equation.

$x = 4$

Step 5

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

$x = - 5$

$x = - 1$

$x = 4$

Step 6

Consolidate the solutions.

$x = - 5, - 1, 4$

Step 7

Find the domain of $\frac{(x + 5) (x + 1)}{x - 4}$ .

Set the denominator in $\frac{(x + 5) (x + 1)}{x - 4}$ equal to $0$ to find where the expression is undefined.

$x - 4 = 0$

Add $4$ to 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 8

Use each root to create test intervals.

$x < - 5$

$- 5 < x < - 1$

$- 1 < x < 4$

$x > 4$

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 $x < - 5$ 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.

$\frac{((- 8) + 5) ((- 8) + 1)}{(- 8) - 4} \geq 0$

The left side $- 1.75$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 5 < x < - 1$ to see if it makes the inequality true.

Choose a value on the interval $- 5 < x < - 1$ and see if this value makes the original inequality true.

$x = - 3$

Replace $x$ with $- 3$ in the original inequality.

$\frac{((- 3) + 5) ((- 3) + 1)}{(- 3) - 4} \geq 0$

The left side $0.\overline{571428}$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 1 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{((0) + 5) ((0) + 1)}{(0) - 4} \geq 0$

The left side $- 1.25$ is less than the right side $0$ , which means that the given statement is false.

False

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.

$\frac{((6) + 5) ((6) + 1)}{(6) - 4} \geq 0$

The left side $38.5$ is greater than the right side $0$ , which means that the given statement is always true.

True

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 5$ False

$- 5 < x < - 1$ True

$- 1 < x < 4$ False

$x > 4$ True

$x < - 5$ False

$- 5 < x < - 1$ True

$- 1 < x < 4$ False

$x > 4$ True

Step 10

The solution consists of all of the true intervals.

$- 5 \leq x \leq - 1$ or $x > 4$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$- 5 \leq x \leq - 1 or x > 4$

Interval Notation:

$[- 5, - 1] \cup (4, \infty)$

Step 12

Solve for x ((x+5)(x+1))/(x-4)>=0

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