Solve the Inequality for x (x-4)(x-6)(x-7)>=0

$(x - 4) (x - 6) (x - 7) \geq 0$

Step 1

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

$x - 4 = 0$

$x - 6 = 0$

$x - 7 = 0$

Step 2

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 3

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

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

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Step 4

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 5

The final solution is all the values that make $(x - 4) (x - 6) (x - 7) \geq 0$ true.

$x = 4, 6, 7$

Step 6

Use each root to create test intervals.

$x < 4$

$4 < x < 6$

$6 < x < 7$

$x > 7$

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.

$x = 0$

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

$((0) - 4) ((0) - 6) ((0) - 7) \geq 0$

The left side $- 168$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $4 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $4 < x < 6$ and see if this value makes the original inequality true.

$x = 5$

Replace $x$ with $5$ in the original inequality.

$((5) - 4) ((5) - 6) ((5) - 7) \geq 0$

The left side $2$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $6 < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $6 < x < 7$ and see if this value makes the original inequality true.

$x = 6.5$

Replace $x$ with $6.5$ in the original inequality.

$((6.5) - 4) ((6.5) - 6) ((6.5) - 7) \geq 0$

The left side $- 0.625$ is 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.

$((10) - 4) ((10) - 6) ((10) - 7) \geq 0$

The left side $72$ 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.

$x < 4$ False

$4 < x < 6$ True

$6 < x < 7$ False

$x > 7$ True

$x < 4$ False

$4 < x < 6$ True

$6 < x < 7$ False

$x > 7$ True

Step 8

The solution consists of all of the true intervals.

$4 \leq x \leq 6$ or $x \geq 7$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$4 \leq x \leq 6 or x \geq 7$

Interval Notation:

$[4, 6] \cup [7, \infty)$

Step 10

Solve the Inequality for x (x-4)(x-6)(x-7)>=0

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