Algebra Examples

$2 x^{2} - 36 x < - 120$

Step 1

Convert the inequality to an equation.

$2 x^{2} - 36 x = - 120$

Step 2

Add $120$ to both sides of the equation.

$2 x^{2} - 36 x + 120 = 0$

Step 3

Factor $2$ out of $2 x^{2} - 36 x + 120$ .

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Step 3.1

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

$2 (x^{2}) - 36 x + 120 = 0$

Step 3.2

Factor $2$ out of $- 36 x$ .

$2 (x^{2}) + 2 (- 18 x) + 120 = 0$

Step 3.3

Factor $2$ out of $120$ .

$2 x^{2} + 2 (- 18 x) + 2 \cdot 60 = 0$

Step 3.4

Factor $2$ out of $2 x^{2} + 2 (- 18 x)$ .

$2 (x^{2} - 18 x) + 2 \cdot 60 = 0$

Step 3.5

Factor $2$ out of $2 (x^{2} - 18 x) + 2 \cdot 60$ .

$2 (x^{2} - 18 x + 60) = 0$

Step 4

Divide each term in $2 (x^{2} - 18 x + 60) = 0$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 (x^{2} - 18 x + 60) = 0$ by $2$ .

$\frac{2 (x^{2} - 18 x + 60)}{2} = \frac{0}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} - 18 x + 60)}{2} = \frac{0}{2}$

Step 4.2.1.2

Divide $x^{2} - 18 x + 60$ by $1$ .

$x^{2} - 18 x + 60 = \frac{0}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $2$ .

$x^{2} - 18 x + 60 = 0$

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values $a = 1$ , $b = - 18$ , and $c = 60$ into the quadratic formula and solve for $x$ .

$\frac{18 \pm \sqrt{{(- 18)}^{2} - 4 \cdot (1 \cdot 60)}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 1 \cdot 60}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 60$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 60}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $60$ .

$x = \frac{18 \pm \sqrt{324 - 240}}{2 \cdot 1}$

Step 7.1.3

Subtract $240$ from $324$ .

$x = \frac{18 \pm \sqrt{84}}{2 \cdot 1}$

Step 7.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 7.1.4.1

Factor $4$ out of $84$ .

$x = \frac{18 \pm \sqrt{4 (21)}}{2 \cdot 1}$

Step 7.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{18 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 1}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{18 \pm 2 \sqrt{21}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{18 \pm 2 \sqrt{21}}{2}$

Step 7.3

Simplify $\frac{18 \pm 2 \sqrt{21}}{2}$ .

$x = 9 \pm \sqrt{21}$

Step 8

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 1 \cdot 60}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot 60$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 60}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $60$ .

$x = \frac{18 \pm \sqrt{324 - 240}}{2 \cdot 1}$

Step 8.1.3

Subtract $240$ from $324$ .

$x = \frac{18 \pm \sqrt{84}}{2 \cdot 1}$

Step 8.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 8.1.4.1

Factor $4$ out of $84$ .

$x = \frac{18 \pm \sqrt{4 (21)}}{2 \cdot 1}$

Step 8.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{18 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 1}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{18 \pm 2 \sqrt{21}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{18 \pm 2 \sqrt{21}}{2}$

Step 8.3

Simplify $\frac{18 \pm 2 \sqrt{21}}{2}$ .

$x = 9 \pm \sqrt{21}$

Step 8.4

Change the $\pm$ to $+$ .

$x = 9 + \sqrt{21}$

Step 9

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 1 \cdot 60}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot 60$ .

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 60}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $60$ .

$x = \frac{18 \pm \sqrt{324 - 240}}{2 \cdot 1}$

Step 9.1.3

Subtract $240$ from $324$ .

$x = \frac{18 \pm \sqrt{84}}{2 \cdot 1}$

Step 9.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 9.1.4.1

Factor $4$ out of $84$ .

$x = \frac{18 \pm \sqrt{4 (21)}}{2 \cdot 1}$

Step 9.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{18 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 1}$

Step 9.1.5

Pull terms out from under the radical.

$x = \frac{18 \pm 2 \sqrt{21}}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$x = \frac{18 \pm 2 \sqrt{21}}{2}$

Step 9.3

Simplify $\frac{18 \pm 2 \sqrt{21}}{2}$ .

$x = 9 \pm \sqrt{21}$

Step 9.4

Change the $\pm$ to $-$ .

$x = 9 - \sqrt{21}$

Step 10

The final answer is the combination of both solutions.

$x = 9 + \sqrt{21}, 9 - \sqrt{21}$

Step 11

Use each root to create test intervals.

$x < 9 - \sqrt{21}$

$9 - \sqrt{21} < x < 9 + \sqrt{21}$

$x > 9 + \sqrt{21}$

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 $x < 9 - \sqrt{21}$ to see if it makes the inequality true.

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Step 12.1.1

Choose a value on the interval $x < 9 - \sqrt{21}$ and see if this value makes the original inequality true.

$x = 0$

Step 12.1.2

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

$2 {(0)}^{2} - 36 \cdot 0 < - 120$

Step 12.1.3

The left side $0$ is not less than the right side $- 120$ , which means that the given statement is false.

False

Step 12.2

Test a value on the interval $9 - \sqrt{21} < x < 9 + \sqrt{21}$ to see if it makes the inequality true.

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Step 12.2.1

Choose a value on the interval $9 - \sqrt{21} < x < 9 + \sqrt{21}$ and see if this value makes the original inequality true.

$x = 9$

Step 12.2.2

Replace $x$ with $9$ in the original inequality.

$2 {(9)}^{2} - 36 \cdot 9 < - 120$

Step 12.2.3

The left side $- 162$ is less than the right side $- 120$ , which means that the given statement is always true.

True

Step 12.3

Test a value on the interval $x > 9 + \sqrt{21}$ to see if it makes the inequality true.

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Step 12.3.1

Choose a value on the interval $x > 9 + \sqrt{21}$ and see if this value makes the original inequality true.

$x = 16$

Step 12.3.2

Replace $x$ with $16$ in the original inequality.

$2 {(16)}^{2} - 36 \cdot 16 < - 120$

Step 12.3.3

The left side $- 64$ is not less than the right side $- 120$ , which means that the given statement is false.

False

Step 12.4

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

$x < 9 - \sqrt{21}$ False

$9 - \sqrt{21} < x < 9 + \sqrt{21}$ True

$x > 9 + \sqrt{21}$ False

$x < 9 - \sqrt{21}$ False

$9 - \sqrt{21} < x < 9 + \sqrt{21}$ True

$x > 9 + \sqrt{21}$ False

Step 13

The solution consists of all of the true intervals.

$9 - \sqrt{21} < x < 9 + \sqrt{21}$

Step 14