Algebra Examples

$X^{3} - 4 X^{2} - X + 4 \geq 0$

Step 1

Convert the inequality to an equation.

$X^{3} - 4 X^{2} - X + 4 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor out the greatest common factor from each group.

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Step 2.1.1

Group the first two terms and the last two terms.

$(X^{3} - 4 X^{2}) - X + 4 = 0$

Step 2.1.2

Factor out the greatest common factor (GCF) from each group.

$X^{2} (X - 4) - (X - 4) = 0$

Step 2.2

Factor the polynomial by factoring out the greatest common factor, $X - 4$ .

$(X - 4) (X^{2} - 1) = 0$

Step 2.3

Rewrite $1$ as $1^{2}$ .

$(X - 4) (X^{2} - 1^{2}) = 0$

Step 2.4

Factor.

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Step 2.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = X$ and $b = 1$ .

$(X - 4) ((X + 1) (X - 1)) = 0$

Step 2.4.2

Remove unnecessary parentheses.

$(X - 4) (X + 1) (X - 1) = 0$

Step 3

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

$X - 4 = 0$

$X + 1 = 0$

$X - 1 = 0$

Step 4

Set $X - 4$ equal to $0$ and solve for $X$ .

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Step 4.1

Set $X - 4$ equal to $0$ .

$X - 4 = 0$

Step 4.2

Add $4$ to both sides of the equation.

$X = 4$

Step 5

Set $X + 1$ equal to $0$ and solve for $X$ .

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Step 5.1

Set $X + 1$ equal to $0$ .

$X + 1 = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$X = - 1$

Step 6

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

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Step 6.1

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

$X - 1 = 0$

Step 6.2

Add $1$ to both sides of the equation.

$X = 1$

Step 7

The final solution is all the values that make $(X - 4) (X + 1) (X - 1) = 0$ true.

$X = 4, - 1, 1$

Step 8

Use each root to create test intervals.

$X < - 1$

$- 1 < X < 1$

$1 < X < 4$

$X > 4$

Step 9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 9.1

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

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Step 9.1.1

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

$X = - 4$

Step 9.1.2

Replace $X$ with $- 4$ in the original inequality.

${(- 4)}^{3} - 4 {(- 4)}^{2} - (- 4) + 4 \geq 0$

Step 9.1.3

The left side $- 120$ is less than the right side $0$ , which means that the given statement is false.

False

Step 9.2

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

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Step 9.2.1

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

$X = 0$

Step 9.2.2

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

${(0)}^{3} - 4 {(0)}^{2} - (0) + 4 \geq 0$

Step 9.2.3

The left side $4$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 9.3

Test a value on the interval $1 < X < 4$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $1 < X < 4$ and see if this value makes the original inequality true.

$X = 2$

Step 9.3.2

Replace $X$ with $2$ in the original inequality.

${(2)}^{3} - 4 {(2)}^{2} - (2) + 4 \geq 0$

Step 9.3.3

The left side $- 6$ is less than the right side $0$ , which means that the given statement is false.

False

Step 9.4

Test a value on the interval $X > 4$ to see if it makes the inequality true.

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Step 9.4.1

Choose a value on the interval $X > 4$ and see if this value makes the original inequality true.

$X = 6$

Step 9.4.2

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

${(6)}^{3} - 4 {(6)}^{2} - (6) + 4 \geq 0$

Step 9.4.3

The left side $70$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 9.5

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

$X < - 1$ False

$- 1 < X < 1$ True

$1 < X < 4$ False

$X > 4$ True

$X < - 1$ False

$- 1 < X < 1$ True

$1 < X < 4$ False

$X > 4$ True

Step 10

The solution consists of all of the true intervals.

$- 1 \leq X \leq 1$ or $X \geq 4$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$- 1 \leq X \leq 1 or X \geq 4$

Interval Notation:

$[- 1, 1] \cup [4, \infty)$

Step 12