Solve the Inequality for x (x+6)(1-x)(x-4)<0

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

Step 1

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

$x + 6 = 0$

$1 - x = 0$

$x - 4 = 0$

Step 2

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 3

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

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

$1 - x = 0$

Solve $1 - x = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$- x = - 1$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 1$ by $- 1$ .

$x = 1$

Step 4

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.

$x = 4$

Step 5

The final solution is all the values that make $(x + 6) (1 - x) (x - 4) < 0$ true.

$x = - 6, 1, 4$

Step 6

Use each root to create test intervals.

$x < - 6$

$- 6 < x < 1$

$1 < x < 4$

$x > 4$

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 $x < - 6$ 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.

$((- 8) + 6) (1 - (- 8)) ((- 8) - 4) < 0$

The left side $216$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 6 < x < 1$ to see if it makes the inequality true.

Choose a value on the interval $- 6 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

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

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

The left side $- 24$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $1 < x < 4$ 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.

$x = 2$

Replace $x$ with $2$ in the original inequality.

$((2) + 6) (1 - (2)) ((2) - 4) < 0$

The left side $16$ is not 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.

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

The left side $- 120$ 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.

$x < - 6$ False

$- 6 < x < 1$ True

$1 < x < 4$ False

$x > 4$ True

$x < - 6$ False

$- 6 < x < 1$ True

$1 < x < 4$ False

$x > 4$ True

Step 8

The solution consists of all of the true intervals.

$- 6 < x < 1$ or $x > 4$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- 6 < x < 1 or x > 4$

Interval Notation:

$(- 6, 1) \cup (4, \infty)$

Step 10

Solve the Inequality for x (x+6)(1-x)(x-4)<0

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