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

$(x - 7) (4 - x) (x + 1) < 0$

Step 1

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

$x - 7 = 0$

$4 - x = 0$

$x + 1 = 0$

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

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

$4 - x = 0$

Solve $4 - x = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

$- x = - 4$

Divide each term in $- x = - 4$ by $- 1$ and simplify.

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Simplify the right side.

Divide $- 4$ by $- 1$ .

$x = 4$

Step 4

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

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

$x = 7, 4, - 1$

Step 6

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 4$

$4 < x < 7$

$x > 7$

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 $x < - 1$ to see if it makes the inequality true.

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

$x = - 4$

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

$((- 4) - 7) (4 - (- 4)) ((- 4) + 1) < 0$

The left side $264$ is not less than the right side $0$ , which means that the given statement is false.

False

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.

$((0) - 7) (4 - (0)) ((0) + 1) < 0$

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 $4 < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $4 < x < 7$ and see if this value makes the original inequality true.

$x = 6$

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

$((6) - 7) (4 - (6)) ((6) + 1) < 0$

The left side $14$ is not less than the right side $0$ , which means that the given statement is false.

False

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.

$((10) - 7) (4 - (10)) ((10) + 1) < 0$

The left side $- 198$ is less 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.

$x < - 1$ False

$- 1 < x < 4$ True

$4 < x < 7$ False

$x > 7$ True

$x < - 1$ False

$- 1 < x < 4$ True

$4 < x < 7$ False

$x > 7$ True

Step 8

The solution consists of all of the true intervals.

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

Step 9

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

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

Step 10

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

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