Algebra Examples

$3 x (x + 1) \leq x (x + 5)$

Step 1

Simplify $3 x (x + 1)$ .

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Step 1.1

Rewrite.

$0 + 0 + 3 x (x + 1) \leq x (x + 5)$

Step 1.2

Simplify by adding zeros.

$3 x (x + 1) \leq x (x + 5)$

Step 1.3

Apply the distributive property.

$3 x \cdot x + 3 x \cdot 1 \leq x (x + 5)$

Step 1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 1.4.1

Move $x$ .

$3 (x \cdot x) + 3 x \cdot 1 \leq x (x + 5)$

Step 1.4.2

Multiply $x$ by $x$ .

$3 x^{2} + 3 x \cdot 1 \leq x (x + 5)$

Step 1.5

Multiply $3$ by $1$ .

$3 x^{2} + 3 x \leq x (x + 5)$

Step 2

Simplify $x (x + 5)$ .

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Step 2.1

Apply the distributive property.

$3 x^{2} + 3 x \leq x \cdot x + x \cdot 5$

Step 2.2

Simplify the expression.

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Step 2.2.1

Multiply $x$ by $x$ .

$3 x^{2} + 3 x \leq x^{2} + x \cdot 5$

Step 2.2.2

Move $5$ to the left of $x$ .

$3 x^{2} + 3 x \leq x^{2} + 5 x$

Step 3

Move all terms containing $x$ to the left side of the inequality.

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Step 3.1

Subtract $x^{2}$ from both sides of the inequality.

$3 x^{2} + 3 x - x^{2} \leq 5 x$

Step 3.2

Subtract $5 x$ from both sides of the inequality.

$3 x^{2} + 3 x - x^{2} - 5 x \leq 0$

Step 3.3

Subtract $x^{2}$ from $3 x^{2}$ .

$2 x^{2} + 3 x - 5 x \leq 0$

Step 3.4

Subtract $5 x$ from $3 x$ .

$2 x^{2} - 2 x \leq 0$

Step 4

Convert the inequality to an equation.

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

Step 5

Factor $2 x$ out of $2 x^{2} - 2 x$ .

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Step 5.1

Factor $2 x$ out of $2 x^{2}$ .

$2 x (x) - 2 x = 0$

Step 5.2

Factor $2 x$ out of $- 2 x$ .

$2 x (x) + 2 x (- 1) = 0$

Step 5.3

Factor $2 x$ out of $2 x (x) + 2 x (- 1)$ .

$2 x (x - 1) = 0$

Step 6

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

$x = 0$

$x - 1 = 0$

Step 7

Set $x$ equal to $0$ .

$x = 0$

Step 8

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

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Step 8.1

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

$x - 1 = 0$

Step 8.2

Add $1$ to both sides of the equation.

$x = 1$

Step 9

The final solution is all the values that make $2 x (x - 1) = 0$ true.

$x = 0, 1$

Step 10

Use each root to create test intervals.

$x < 0$

$0 < x < 1$

$x > 1$

Step 11

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 11.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 11.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 11.1.2

Replace $x$ with $- 2$ in the original inequality.

$3 (- 2) ((- 2) + 1) \leq (- 2) ((- 2) + 5)$

Step 11.1.3

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

False

Step 11.2

Test a value on the interval $0 < x < 1$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $0 < x < 1$ and see if this value makes the original inequality true.

$x = 0.5$

Step 11.2.2

Replace $x$ with $0.5$ in the original inequality.

$3 (0.5) ((0.5) + 1) \leq (0.5) ((0.5) + 5)$

Step 11.2.3

The left side $2.25$ is less than the right side $2.75$ , which means that the given statement is always true.

True

Step 11.3

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

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Step 11.3.1

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

$x = 4$

Step 11.3.2

Replace $x$ with $4$ in the original inequality.

$3 (4) ((4) + 1) \leq (4) ((4) + 5)$

Step 11.3.3

The left side $60$ is greater than the right side $36$ , which means that the given statement is false.

False

Step 11.4

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

$x < 0$ False

$0 < x < 1$ True

$x > 1$ False

$x < 0$ False

$0 < x < 1$ True

$x > 1$ False

Step 12

The solution consists of all of the true intervals.

$0 \leq x \leq 1$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$0 \leq x \leq 1$

Interval Notation:

$[0, 1]$

Step 14