Solve the Inequality for x x(x-1)<=20

$x (x - 1) \leq 20$

Step 1

Simplify $x (x - 1)$ .

Simplify by multiplying through.

Apply the distributive property.

$x \cdot x + x \cdot - 1 \leq 20$

Simplify the expression.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 \leq 20$

Move $- 1$ to the left of $x$ .

$x^{2} - 1 \cdot x \leq 20$

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x \leq 20$

Step 2

Convert the inequality to an equation.

$x^{2} - x = 20$

Step 3

Subtract $20$ from both sides of the equation.

$x^{2} - x - 20 = 0$

Step 4

Factor $x^{2} - x - 20$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Write the factored form using these integers.

$(x - 5) (x + 4) = 0$

Step 5

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

$x - 5 = 0$

$x + 4 = 0$

Step 6

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 7

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 8

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

$x = 5, - 4$

Step 9

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 5$

$x > 5$

Step 10

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.

$x = - 6$

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

$(- 6) ((- 6) - 1) \leq 20$

The left side $42$ is greater than the right side $20$ , 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.

$x = 0$

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

$(0) ((0) - 1) \leq 20$

The left side $0$ is less than the right side $20$ , 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.

$(8) ((8) - 1) \leq 20$

The left side $56$ is greater than the right side $20$ , which means that the given statement is false.

False

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 4$ False

$- 4 < x < 5$ True

$x > 5$ False

$x < - 4$ False

$- 4 < x < 5$ True

$x > 5$ False

Step 11

The solution consists of all of the true intervals.

$- 4 \leq x \leq 5$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$- 4 \leq x \leq 5$

Interval Notation:

$[- 4, 5]$

Step 13

Solve the Inequality for x x(x-1)<=20

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