Solve the Inequality for x (x-7)(x-1)^2(x+4)<0

$(x - 7) {(x - 1)}^{2} (x + 4) < 0$

Step 1

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

$x - 7 = 0$

${(x - 1)}^{2} = 0$

$x + 4 = 0$

Step 2

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Step 3

Set ${(x - 1)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 1)}^{2}$ equal to $0$ .

${(x - 1)}^{2} = 0$

Solve ${(x - 1)}^{2} = 0$ for $x$ .

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Step 4

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

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

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 5

The final solution is all the values that make $(x - 7) {(x - 1)}^{2} (x + 4) < 0$ true.

$x = 7, 1, - 4$

Step 6

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 1$

$1 < x < 7$

$x > 7$

Step 7

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 $x < - 4$ 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.

$((- 6) - 7) {((- 6) - 1)}^{2} ((- 6) + 4) < 0$

The left side $1274$ is not less than the right side $0$ , which means that the given statement is false.

False

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

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

$x = 0$

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

$((0) - 7) {((0) - 1)}^{2} ((0) + 4) < 0$

The left side $- 28$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $1 < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $1 < x < 7$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

$((4) - 7) {((4) - 1)}^{2} ((4) + 4) < 0$

The left side $- 216$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 7$ to see if it makes the inequality true.

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Replace $x$ with $10$ in the original inequality.

$((10) - 7) {((10) - 1)}^{2} ((10) + 4) < 0$

The left side $3402$ 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.

$x < - 4$ False

$- 4 < x < 1$ True

$1 < x < 7$ True

$x > 7$ False

$x < - 4$ False

$- 4 < x < 1$ True

$1 < x < 7$ True

$x > 7$ False

Step 8

The solution consists of all of the true intervals.

$- 4 < x < 1$ or $1 < x < 7$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- 4 < x < 1 or 1 < x < 7$

Interval Notation:

$(- 4, 1) \cup (1, 7)$

Step 10

Solve the Inequality for x (x-7)(x-1)^2(x+4)<0

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