Solve the Inequality for x (x-2)(x+4)(x+7)>0

$(x - 2) (x + 4) (x + 7) > 0$

Step 1

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

$x - 2 = 0$

$x + 4 = 0$

$x + 7 = 0$

Step 2

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 3

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 4

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

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

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 5

The final solution is all the values that make $(x - 2) (x + 4) (x + 7) > 0$ true.

$x = 2, - 4, - 7$

Step 6

Use each root to create test intervals.

$x < - 7$

$- 7 < x < - 4$

$- 4 < x < 2$

$x > 2$

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 $x < - 7$ 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.

$((- 10) - 2) ((- 10) + 4) ((- 10) + 7) > 0$

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

False

Test a value on the interval $- 7 < x < - 4$ to see if it makes the inequality true.

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

$x = - 6$

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

$((- 6) - 2) ((- 6) + 4) ((- 6) + 7) > 0$

The left side $16$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 4 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

$((0) - 2) ((0) + 4) ((0) + 7) > 0$

The left side $- 56$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 2$ to see if it makes the inequality true.

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

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

$((4) - 2) ((4) + 4) ((4) + 7) > 0$

The left side $176$ is greater 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.

$x < - 7$ False

$- 7 < x < - 4$ True

$- 4 < x < 2$ False

$x > 2$ True

$x < - 7$ False

$- 7 < x < - 4$ True

$- 4 < x < 2$ False

$x > 2$ True

Step 8

The solution consists of all of the true intervals.

$- 7 < x < - 4$ or $x > 2$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- 7 < x < - 4 or x > 2$

Interval Notation:

$(- 7, - 4) \cup (2, \infty)$

Step 10

Solve the Inequality for x (x-2)(x+4)(x+7)>0

Related Questions

Support LCMGCF.com by disabling your adblocker.