Solve the Inequality for x ((x-3)(x-4))/((x-5)(x-6)^2)<0

$\frac{(x - 3) (x - 4)}{(x - 5) {(x - 6)}^{2}} < 0$

Step 1

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

$x - 3 = 0$

$x - 4 = 0$

$x - 5 = 0$

${(x - 6)}^{2} = 0$

Step 2

Add $3$ to both sides of the equation.

$x = 3$

Step 3

Add $4$ to both sides of the equation.

$x = 4$

Step 4

Add $5$ to both sides of the equation.

$x = 5$

Step 5

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 6

Add $6$ to both sides of the equation.

$x = 6$

Step 7

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

$x = 3$

$x = 4$

$x = 5$

$x = 6$

Step 8

Consolidate the solutions.

$x = 3, 4, 5, 6$

Step 9

Find the domain of $\frac{(x - 3) (x - 4)}{(x - 5) {(x - 6)}^{2}}$ .

Set the denominator in $\frac{(x - 3) (x - 4)}{(x - 5) {(x - 6)}^{2}}$ equal to $0$ to find where the expression is undefined.

$(x - 5) {(x - 6)}^{2} = 0$

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 $0$ .

$x - 5 = 0$

${(x - 6)}^{2} = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set ${(x - 6)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 6)}^{2}$ equal to $0$ .

${(x - 6)}^{2} = 0$

Solve ${(x - 6)}^{2} = 0$ for $x$ .

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

The final solution is all the values that make $(x - 5) {(x - 6)}^{2} = 0$ true.

$x = 5, 6$

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

$(- \infty, 5) \cup (5, 6) \cup (6, \infty)$

Step 10

Use each root to create test intervals.

$x < 3$

$3 < x < 4$

$4 < x < 5$

$5 < x < 6$

$x > 6$

Step 11

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.

$x = 0$

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

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

The left side $- 0.0\overline{6}$ is less than the right side $0$ , which means that the given statement is always true.

True

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.

$x = 3.5$

Replace $x$ with $3.5$ in the original inequality.

$\frac{((3.5) - 3) ((3.5) - 4)}{((3.5) - 5) {((3.5) - 6)}^{2}} < 0$

The left side $0.02\overline{6}$ is not less 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.

$\frac{((4.5) - 3) ((4.5) - 4)}{((4.5) - 5) {((4.5) - 6)}^{2}} < 0$

The left side $- 0.\overline{6}$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $5 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $5 < x < 6$ and see if this value makes the original inequality true.

$x = 5.5$

Replace $x$ with $5.5$ in the original inequality.

$\frac{((5.5) - 3) ((5.5) - 4)}{((5.5) - 5) {((5.5) - 6)}^{2}} < 0$

The left side $30$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 6$ to see if it makes the inequality true.

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

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

$\frac{((8) - 3) ((8) - 4)}{((8) - 5) {((8) - 6)}^{2}} < 0$

The left side $1.\overline{6}$ is not 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.

$x < 3$ True

$3 < x < 4$ False

$4 < x < 5$ True

$5 < x < 6$ False

$x > 6$ False

$x < 3$ True

$3 < x < 4$ False

$4 < x < 5$ True

$5 < x < 6$ False

$x > 6$ False

Step 12

The solution consists of all of the true intervals.

$x < 3$ or $4 < x < 5$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$x < 3 or 4 < x < 5$

Interval Notation:

$(- \infty, 3) \cup (4, 5)$

Step 14

Solve the Inequality for x ((x-3)(x-4))/((x-5)(x-6)^2)<0

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