Solve the Inequality for x (x-5)/((x+3)^2(x-4))<=0

$\frac{x - 5}{{(x + 3)}^{2} (x - 4)} \leq 0$

Step 1

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

$x - 5 = 0$

${(x + 3)}^{2} = 0$

$x - 4 = 0$

Step 2

Add $5$ to both sides of the equation.

$x = 5$

Step 3

Set the $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 4

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 5

Add $4$ to both sides of the equation.

$x = 4$

Step 6

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

$x = 5$

$x = - 3$

$x = 4$

Step 7

Consolidate the solutions.

$x = 5, - 3, 4$

Step 8

Find the domain of $\frac{x - 5}{{(x + 3)}^{2} (x - 4)}$ .

Set the denominator in $\frac{x - 5}{{(x + 3)}^{2} (x - 4)}$ equal to $0$ to find where the expression is undefined.

${(x + 3)}^{2} (x - 4) = 0$

Solve for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x + 3)}^{2} = 0$

$x - 4 = 0$

Set ${(x + 3)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x + 3)}^{2}$ equal to $0$ .

${(x + 3)}^{2} = 0$

Solve ${(x + 3)}^{2} = 0$ for $x$ .

Set the $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make ${(x + 3)}^{2} (x - 4) = 0$ true.

$x = - 3, 4$

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

$(- \infty, - 3) \cup (- 3, 4) \cup (4, \infty)$

Step 9

Use each root to create test intervals.

$x < - 3$

$- 3 < x < 4$

$4 < x < 5$

$x > 5$

Step 10

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 $x < - 3$ to see if it makes the inequality true.

Choose a value on the interval $x < - 3$ and see if this value makes the original inequality true.

$x = - 6$

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

$\frac{(- 6) - 5}{{((- 6) + 3)}^{2} ((- 6) - 4)} \leq 0$

The left side $0.1\overline{2}$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 3 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $- 3 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{(0) - 5}{{((0) + 3)}^{2} ((0) - 4)} \leq 0$

The left side $0.13\overline{8}$ is greater than the right side $0$ , which means that the given statement is false.

False

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

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

$x = 4.5$

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

$\frac{(4.5) - 5}{{((4.5) + 3)}^{2} ((4.5) - 4)} \leq 0$

The left side $- 0.01\overline{7}$ is less than the right side $0$ , which means that the given statement is always true.

True

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.

$\frac{(8) - 5}{{((8) + 3)}^{2} ((8) - 4)} \leq 0$

The left side $0.00619834$ 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.

$x < - 3$ False

$- 3 < x < 4$ False

$4 < x < 5$ True

$x > 5$ False

$x < - 3$ False

$- 3 < x < 4$ False

$4 < x < 5$ True

$x > 5$ False

Step 11

The solution consists of all of the true intervals.

$4 < x \leq 5$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$4 < x \leq 5$

Interval Notation:

$(4, 5]$

Step 13

Solve the Inequality for x (x-5)/((x+3)^2(x-4))<=0

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