Solve the Inequality for x (x-2)(x-3)(x-4)<=0

$(x - 2) (x - 3) (x - 4) \leq 0$

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

$x - 2 = 0$

$x - 3 = 0$

$x - 4 = 0$

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.

$x = 2$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

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.

$x = 4$

The final solution is all the values that make $(x - 2) (x - 3) (x - 4) \leq 0$ true.

$x = 2, 3, 4$

Use each root to create test intervals.

$x < 2$

$2 < x < 3$

$3 < x < 4$

$x > 4$

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 $x < 2$ 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.

$x = 0$

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

$((0) - 2) ((0) - 3) ((0) - 4) \leq 0$

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 $2 < x < 3$ to see if it makes the inequality true.

Choose a value on the interval $2 < x < 3$ and see if this value makes the original inequality true.

$x = 2.5$

Replace $x$ with $2.5$ in the original inequality.

$((2.5) - 2) ((2.5) - 3) ((2.5) - 4) \leq 0$

The left side $0.375$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $3 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $3 < x < 4$ and see if this value makes the original inequality true.

$x = 3.5$

Replace $x$ with $3.5$ in the original inequality.

$((3.5) - 2) ((3.5) - 3) ((3.5) - 4) \leq 0$

The left side $- 0.375$ is less than the right side $0$ , which means that the given statement is always true.

True

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.

$((6) - 2) ((6) - 3) ((6) - 4) \leq 0$

The left side $24$ 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.

$x < 2$ True

$2 < x < 3$ False

$3 < x < 4$ True

$x > 4$ False

$x < 2$ True

$2 < x < 3$ False

$3 < x < 4$ True

$x > 4$ False

The solution consists of all of the true intervals.

$x \leq 2$ or $3 \leq x \leq 4$

The result can be shown in multiple forms.

Inequality Form:

$x \leq 2 or 3 \leq x \leq 4$

Interval Notation:

$(- \infty, 2] \cup [3, 4]$

Solve the Inequality for x (x-2)(x-3)(x-4)<=0

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