Simplify (x-3)(x-4)(x-7)<=0

$(x - 3) (x - 4) (x - 7) \leq 0$

Step 1

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

$x - 3 = 0$

$x - 4 = 0$

$x - 7 = 0$

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 $x - 7$ 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 $(x - 3) (x - 4) (x - 7) \leq 0$ true.

$x = 3, 4, 7$

Step 6

Use each root to create test intervals.

$x < 3$

$3 < x < 4$

$4 < x < 7$

$x > 7$

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 $x < 3$ 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.

$((0) - 3) ((0) - 4) ((0) - 7) \leq 0$

The left side $- 84$ 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.

$((3.5) - 3) ((3.5) - 4) ((3.5) - 7) \leq 0$

The left side $0.875$ is greater than the right side $0$ , which means that the given statement is false.

False

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

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

$x = 6$

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

$((6) - 3) ((6) - 4) ((6) - 7) \leq 0$

The left side $- 6$ 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.

$((10) - 3) ((10) - 4) ((10) - 7) \leq 0$

The left side $126$ 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

$4 < x < 7$ True

$x > 7$ False

$x < 3$ True

$3 < x < 4$ False

$4 < x < 7$ True

$x > 7$ False

Step 8

The solution consists of all of the true intervals.

$x \leq 3$ or $4 \leq x \leq 7$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$x \leq 3 or 4 \leq x \leq 7$

Interval Notation:

$(- \infty, 3] \cup [4, 7]$

Step 10

Simplify (x-3)(x-4)(x-7)<=0

Related Questions