Algebra Examples

$(3 x + 10) (2 x^{2} + 1) (2 - x) > 0$

Step 1

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

$3 x + 10 = 0$

$2 x^{2} + 1 = 0$

$2 - x = 0$

Step 2

Set $3 x + 10$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $3 x + 10$ equal to $0$ .

$3 x + 10 = 0$

Step 2.2

Solve $3 x + 10 = 0$ for $x$ .

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Step 2.2.1

Subtract $10$ from both sides of the equation.

$3 x = - 10$

Step 2.2.2

Divide each term in $3 x = - 10$ by $3$ and simplify.

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Step 2.2.2.1

Divide each term in $3 x = - 10$ by $3$ .

$\frac{3 x}{3} = \frac{- 10}{3}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 10}{3}$

Step 2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 10}{3}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{10}{3}$

Step 3

Set $2 x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $2 x^{2} + 1$ equal to $0$ .

$2 x^{2} + 1 = 0$

Step 3.2

Solve $2 x^{2} + 1 = 0$ for $x$ .

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$2 x^{2} = - 1$

Step 3.2.2

Divide each term in $2 x^{2} = - 1$ by $2$ and simplify.

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Step 3.2.2.1

Divide each term in $2 x^{2} = - 1$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{- 1}{2}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{- 1}{2}$

Step 3.2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 1}{2}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{1}{2}$

Step 3.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- \frac{1}{2}}$

Step 3.2.4

Simplify $\pm \sqrt{- \frac{1}{2}}$ .

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Step 3.2.4.1

Rewrite $- \frac{1}{2}$ as $i^{2} \frac{1^{2}}{2}$ .

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Step 3.2.4.1.1

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

$x = \pm \sqrt{i^{2} \frac{1}{2}}$

Step 3.2.4.1.2

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

$x = \pm \sqrt{i^{2} \frac{1^{2}}{2}}$

Step 3.2.4.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{1^{2}}{2}}$

Step 3.2.4.3

One to any power is one.

$x = \pm i \sqrt{\frac{1}{2}}$

Step 3.2.4.4

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{1}}{\sqrt{2}}$

Step 3.2.4.5

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{2}}$

Step 3.2.4.6

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 3.2.4.7

Combine and simplify the denominator.

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Step 3.2.4.7.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 3.2.4.7.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.2.4.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.2.4.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.2.4.7.5

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.2.4.7.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.2.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.2.4.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.4.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm i \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.2.4.7.6.4

Cancel the common factor of $2$ .

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Step 3.2.4.7.6.4.1

Cancel the common factor.

$x = \pm i \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.2.4.7.6.4.2

Rewrite the expression.

$x = \pm i \frac{\sqrt{2}}{2^{1}}$

Step 3.2.4.7.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{2}}{2}$

Step 3.2.4.8

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

$x = \pm \frac{i \sqrt{2}}{2}$

Step 3.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{i \sqrt{2}}{2}$

Step 3.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{i \sqrt{2}}{2}$

Step 3.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{i \sqrt{2}}{2}, - \frac{i \sqrt{2}}{2}$

Step 4

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

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Step 4.1

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

$2 - x = 0$

Step 4.2

Solve $2 - x = 0$ for $x$ .

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Step 4.2.1

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 4.2.2

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 4.2.2.1

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Step 4.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 5

The final solution is all the values that make $(3 x + 10) (2 x^{2} + 1) (2 - x) > 0$ true.

$x = - \frac{10}{3}, \frac{i \sqrt{2}}{2}, - \frac{i \sqrt{2}}{2}, 2$

Step 6

Use each root to create test intervals.

$x < - \frac{10}{3}$

$- \frac{10}{3} < x < 2$

$x > 2$

Step 7

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 7.1

Test a value on the interval $x < - \frac{10}{3}$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $x < - \frac{10}{3}$ and see if this value makes the original inequality true.

$x = - 6$

Step 7.1.2

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

$(3 (- 6) + 10) (2 {(- 6)}^{2} + 1) (2 - (- 6)) > 0$

Step 7.1.3

The left side $- 4672$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 7.2

Test a value on the interval $- \frac{10}{3} < x < 2$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $- \frac{10}{3} < x < 2$ and see if this value makes the original inequality true.

$x = 0$

Step 7.2.2

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

$(3 (0) + 10) (2 {(0)}^{2} + 1) (2 - (0)) > 0$

Step 7.2.3

The left side $20$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.3

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

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Step 7.3.1

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

$x = 4$

Step 7.3.2

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

$(3 (4) + 10) (2 {(4)}^{2} + 1) (2 - (4)) > 0$

Step 7.3.3

The left side $- 1452$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 7.4

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

$x < - \frac{10}{3}$ False

$- \frac{10}{3} < x < 2$ True

$x > 2$ False

$x < - \frac{10}{3}$ False

$- \frac{10}{3} < x < 2$ True

$x > 2$ False

Step 8

The solution consists of all of the true intervals.

$- \frac{10}{3} < x < 2$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- \frac{10}{3} < x < 2$

Interval Notation:

$(- \frac{10}{3}, 2)$

Step 10