Solve the Inequality for x (x-3)(x-4)(x-5)<=0

$(x - 3) (x - 4) (x - 5) \leq 0$

Step 1

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

$x - 3 = 0$

$x - 4 = 0$

$x - 5 = 0$

Step 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.

$x = 3$

Step 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.

$x = 4$

Step 4

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Step 5

The final solution is all the values that make $(x - 3) (x - 4) (x - 5) \leq 0$ true.

$x = 3, 4, 5$

Step 6

Use each root to create test intervals.

$x < 3$

$3 < x < 4$

$4 < x < 5$

$x > 5$

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

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

$x = 0$

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

$((0) - 3) ((0) - 4) ((0) - 5) \leq 0$

The left side $- 60$ is less than the right side $0$ , which means that the given statement is always true.

True

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.

$((3.5) - 3) ((3.5) - 4) ((3.5) - 5) \leq 0$

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 $4 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $4 < x < 5$ and see if this value makes the original inequality true.

$x = 4.5$

Replace $x$ with $4.5$ in the original inequality.

$((4.5) - 3) ((4.5) - 4) ((4.5) - 5) \leq 0$

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 $x > 5$ to see if it makes the inequality true.

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Replace $x$ with $8$ in the original inequality.

$((8) - 3) ((8) - 4) ((8) - 5) \leq 0$

The left side $60$ 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.

$x < 3$ True

$3 < x < 4$ False

$4 < x < 5$ True

$x > 5$ False

$x < 3$ True

$3 < x < 4$ False

$4 < x < 5$ True

$x > 5$ False

Step 8

The solution consists of all of the true intervals.

$x \leq 3$ or $4 \leq x \leq 5$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$x \leq 3 or 4 \leq x \leq 5$

Interval Notation:

$(- \infty, 3] \cup [4, 5]$

Step 10

Solve the Inequality for x (x-3)(x-4)(x-5)<=0

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