Algebra Examples

$x (3 x + 2) > {(x + 2)}^{2}$

Step 1

Simplify $x (3 x + 2)$ .

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Step 1.1

Rewrite.

$0 + 0 + x (3 x + 2) > {(x + 2)}^{2}$

Step 1.2

Simplify by multiplying through.

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Step 1.2.1

Apply the distributive property.

$x (3 x) + x \cdot 2 > {(x + 2)}^{2}$

Step 1.2.2

Reorder.

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Step 1.2.2.1

Rewrite using the commutative property of multiplication.

$3 x \cdot x + x \cdot 2 > {(x + 2)}^{2}$

Step 1.2.2.2

Move $2$ to the left of $x$ .

$3 x \cdot x + 2 \cdot x > {(x + 2)}^{2}$

Step 1.3

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

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Step 1.3.1

Move $x$ .

$3 (x \cdot x) + 2 \cdot x > {(x + 2)}^{2}$

Step 1.3.2

Multiply $x$ by $x$ .

$3 x^{2} + 2 \cdot x > {(x + 2)}^{2}$

$3 x^{2} + 2 x > {(x + 2)}^{2}$

Step 2

Simplify ${(x + 2)}^{2}$ .

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Step 2.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$3 x^{2} + 2 x > (x + 2) (x + 2)$

Step 2.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$3 x^{2} + 2 x > x (x + 2) + 2 (x + 2)$

Step 2.2.2

Apply the distributive property.

$3 x^{2} + 2 x > x \cdot x + x \cdot 2 + 2 (x + 2)$

Step 2.2.3

Apply the distributive property.

$3 x^{2} + 2 x > x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $x$ by $x$ .

$3 x^{2} + 2 x > x^{2} + x \cdot 2 + 2 x + 2 \cdot 2$

Step 2.3.1.2

Move $2$ to the left of $x$ .

$3 x^{2} + 2 x > x^{2} + 2 \cdot x + 2 x + 2 \cdot 2$

Step 2.3.1.3

Multiply $2$ by $2$ .

$3 x^{2} + 2 x > x^{2} + 2 x + 2 x + 4$

Step 2.3.2

Add $2 x$ and $2 x$ .

$3 x^{2} + 2 x > x^{2} + 4 x + 4$

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.

$3 x^{2} + 2 x - x^{2} > 4 x + 4$

Step 3.2

Subtract $4 x$ from both sides of the inequality.

$3 x^{2} + 2 x - x^{2} - 4 x > 4$

Step 3.3

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

$2 x^{2} + 2 x - 4 x > 4$

Step 3.4

Subtract $4 x$ from $2 x$ .

$2 x^{2} - 2 x > 4$

Step 4

Convert the inequality to an equation.

$2 x^{2} - 2 x = 4$

Step 5

Subtract $4$ from both sides of the equation.

$2 x^{2} - 2 x - 4 = 0$

Step 6

Factor the left side of the equation.

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Step 6.1

Factor $2$ out of $2 x^{2} - 2 x - 4$ .

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

Factor $2$ out of $- 4$ .

$2 (x^{2}) + 2 (- x) + 2 (- 2) = 0$

Step 6.1.4

Factor $2$ out of $2 (x^{2}) + 2 (- x)$ .

$2 (x^{2} - x) + 2 (- 2) = 0$

Step 6.1.5

Factor $2$ out of $2 (x^{2} - x) + 2 (- 2)$ .

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

Step 6.2

Factor.

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Step 6.2.1

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

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Step 6.2.1.1

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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 6.2.1.2

Write the factored form using these integers.

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

Step 6.2.2

Remove unnecessary parentheses.

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

Step 7

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

$x - 2 = 0$

$x + 1 = 0$

Step 8

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 8.2

Add $2$ to both sides of the equation.

$x = 2$

Step 9

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

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Step 9.1

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

$x + 1 = 0$

Step 9.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 10

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

$x = 2, - 1$

Step 11

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 2$

$x > 2$

Step 12

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 12.1

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

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Step 12.1.1

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

$x = - 4$

Step 12.1.2

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

$(- 4) (3 (- 4) + 2) > {((- 4) + 2)}^{2}$

Step 12.1.3

The left side $40$ is greater than the right side $4$ , which means that the given statement is always true.

True

Step 12.2

Test a value on the interval $- 1 < x < 2$ to see if it makes the inequality true.

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Step 12.2.1

Choose a value on the interval $- 1 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

Step 12.2.2

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

$(0) (3 (0) + 2) > {((0) + 2)}^{2}$

Step 12.2.3

The left side $0$ is not greater than the right side $4$ , which means that the given statement is false.

False

Step 12.3

Test a value on the interval $x > 2$ to see if it makes the inequality true.

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Step 12.3.1

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Step 12.3.2

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

$(4) (3 (4) + 2) > {((4) + 2)}^{2}$

Step 12.3.3

The left side $56$ is greater than the right side $36$ , which means that the given statement is always true.

True

Step 12.4

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

$x < - 1$ True

$- 1 < x < 2$ False

$x > 2$ True

$x < - 1$ True

$- 1 < x < 2$ False

$x > 2$ True

Step 13

The solution consists of all of the true intervals.

$x < - 1$ or $x > 2$

Step 14

The result can be shown in multiple forms.

Inequality Form:

$x < - 1 or x > 2$

Interval Notation:

$(- \infty, - 1) \cup (2, \infty)$

Step 15