Precalculus Examples

$x^{3} - 1 = | x + 1 |$

Step 1

Rewrite the equation as $| x + 1 | = x^{3} - 1$ .

$| x + 1 | = x^{3} - 1$

Step 2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x + 1 = \pm (x^{3} - 1)$

Step 3

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

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Step 3.1

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

$x + 1 = x^{3} - 1$

Step 3.2

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

$x + 1 = - (x^{3} - 1)$

Step 3.3

Simplify $- (x^{3} - 1)$ .

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Step 3.3.1

Rewrite.

$x + 1 = 0 + 0 - (x^{3} - 1)$

Step 3.3.2

Simplify by adding zeros.

$x + 1 = - (x^{3} - 1)$

Step 3.3.3

Apply the distributive property.

$x + 1 = - x^{3} - - 1$

Step 3.3.4

Multiply $- 1$ by $- 1$ .

$x + 1 = - x^{3} + 1$

Step 3.4

Add $x^{3}$ to both sides of the equation.

$x + 1 + x^{3} = 1$

Step 3.5

Subtract $1$ from both sides of the equation.

$x + 1 + x^{3} - 1 = 0$

Step 3.6

Combine the opposite terms in $x + 1 + x^{3} - 1$ .

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Step 3.6.1

Subtract $1$ from $1$ .

$x + x^{3} + 0 = 0$

Step 3.6.2

Add $x + x^{3}$ and $0$ .

$x + x^{3} = 0$

Step 3.7

Factor $x$ out of $x + x^{3}$ .

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Step 3.7.1

Factor $x$ out of $x^{1}$ .

$x \cdot 1 + x^{3} = 0$

Step 3.7.2

Factor $x$ out of $x^{3}$ .

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

Step 3.7.3

Factor $x$ out of $x \cdot 1 + x \cdot x^{2}$ .

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

Step 3.8

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

$x = 0$

$1 + x^{2} = 0$

Step 3.9

Set $x$ equal to $0$ .

$x = 0$

Step 3.10

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

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Step 3.10.1

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

$1 + x^{2} = 0$

Step 3.10.2

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

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Step 3.10.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 3.10.2.2

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

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

Step 3.10.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 3.10.2.4

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

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Step 3.10.2.4.1

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

$x = i$

Step 3.10.2.4.2

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

$x = - i$

Step 3.10.2.4.3

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

$x = i, - i$

Step 3.11

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

$x = 0, i, - i$