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

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

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

$x - 4 = 0$

$x - 6 = 0$

$x - 7 = 0$

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.

$x = 4$

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.

$x = 6$

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.

$x = 7$

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

$x = 4, 6, 7$

Use each root to create test intervals.

$x < 4$

$4 < x < 6$

$6 < x < 7$

$x > 7$

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.

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

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

True

Test a value on the interval $4 < x < 6$ 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.

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

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

False

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.

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

The left side $- 0.625$ 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.

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

The left side $72$ is greater 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.

$x < 4$ True

$4 < x < 6$ False

$6 < x < 7$ True

$x > 7$ False

$x < 4$ True

$4 < x < 6$ False

$6 < x < 7$ True

$x > 7$ False

The solution consists of all of the true intervals.

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

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(- \infty, 4] \cup [6, 7]$

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

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