Solve the Inequality for x 4(x-1)(x-5)>0

$4 (x - 1) (x - 5) > 0$

Step 1

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

$x - 1 = 0$

$x - 5 = 0$

Step 2

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Step 3

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 4

The final solution is all the values that make $4 (x - 1) (x - 5) > 0$ true.

$x = 1, 5$

Step 5

Use each root to create test intervals.

$x < 1$

$1 < x < 5$

$x > 5$

Step 6

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 $x < 1$ to see if it makes the inequality true.

Choose a value on the interval $x < 1$ and see if this value makes the original inequality true.

$x = 0$

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

$4 ((0) - 1) ((0) - 5) > 0$

The left side $20$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $1 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $1 < x < 5$ and see if this value makes the original inequality true.

$x = 3$

Replace $x$ with $3$ in the original inequality.

$4 ((3) - 1) ((3) - 5) > 0$

The left side $- 16$ is not greater than the right side $0$ , which means that the given statement is false.

False

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.

$4 ((8) - 1) ((8) - 5) > 0$

The left side $84$ 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.

$x < 1$ True

$1 < x < 5$ False

$x > 5$ True

$x < 1$ True

$1 < x < 5$ False

$x > 5$ True

Step 7

The solution consists of all of the true intervals.

$x < 1$ or $x > 5$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x < 1 or x > 5$

Interval Notation:

$(- \infty, 1) \cup (5, \infty)$

Step 9

Solve the Inequality for x 4(x-1)(x-5)>0

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