Calculus Examples

$| \frac{1 - 2 x}{x} | = 3 x$

Step 1

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

$\frac{1 - 2 x}{x} = \pm 3 x$

Step 2

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

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Step 2.1

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

$\frac{1 - 2 x}{x} = 3 x$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$x$

Step 2.3

Multiply each term in $\frac{1 - 2 x}{x} = 3 x$ by $x$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{1 - 2 x}{x} = 3 x$ by $x$ .

$\frac{1 - 2 x}{x} x = 3 x \cdot x$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $x$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{1 - 2 x}{x} x = 3 x \cdot x$

Step 2.3.2.1.2

Rewrite the expression.

$1 - 2 x = 3 x \cdot x$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

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

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Step 2.3.3.1.1

Move $x$ .

$1 - 2 x = 3 (x \cdot x)$

Step 2.3.3.1.2

Multiply $x$ by $x$ .

$1 - 2 x = 3 x^{2}$

Step 2.4

Solve the equation.

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Step 2.4.1

Subtract $3 x^{2}$ from both sides of the equation.

$1 - 2 x - 3 x^{2} = 0$

Step 2.4.2

Factor the left side of the equation.

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Step 2.4.2.1

Factor $- 1$ out of $1 - 2 x - 3 x^{2}$ .

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Step 2.4.2.1.1

Reorder the expression.

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Step 2.4.2.1.1.1

Move $1$ .

$- 2 x - 3 x^{2} + 1 = 0$

Step 2.4.2.1.1.2

Reorder $- 2 x$ and $- 3 x^{2}$ .

$- 3 x^{2} - 2 x + 1 = 0$

Step 2.4.2.1.2

Factor $- 1$ out of $- 3 x^{2}$ .

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

Step 2.4.2.1.3

Factor $- 1$ out of $- 2 x$ .

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

Step 2.4.2.1.4

Rewrite $1$ as $- 1 (- 1)$ .

$- (3 x^{2}) - (2 x) - 1 \cdot - 1 = 0$

Step 2.4.2.1.5

Factor $- 1$ out of $- (3 x^{2}) - (2 x)$ .

$- (3 x^{2} + 2 x) - 1 \cdot - 1 = 0$

Step 2.4.2.1.6

Factor $- 1$ out of $- (3 x^{2} + 2 x) - 1 (- 1)$ .

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

Step 2.4.2.2

Factor.

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Step 2.4.2.2.1

Factor by grouping.

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Step 2.4.2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 1 = - 3$ and whose sum is $b = 2$ .

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Step 2.4.2.2.1.1.1

Factor $2$ out of $2 x$ .

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

Step 2.4.2.2.1.1.2

Rewrite $2$ as $- 1$ plus $3$

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

Step 2.4.2.2.1.1.3

Apply the distributive property.

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

Step 2.4.2.2.1.2

Factor out the greatest common factor from each group.

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Step 2.4.2.2.1.2.1

Group the first two terms and the last two terms.

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

Step 2.4.2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- (x (3 x - 1) + 1 (3 x - 1)) = 0$

Step 2.4.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$- ((3 x - 1) (x + 1)) = 0$

Step 2.4.2.2.2

Remove unnecessary parentheses.

$- (3 x - 1) (x + 1) = 0$

Step 2.4.3

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

$3 x - 1 = 0$

$x + 1 = 0$

Step 2.4.4

Set $3 x - 1$ equal to $0$ and solve for $x$ .

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Step 2.4.4.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 2.4.4.2

Solve $3 x - 1 = 0$ for $x$ .

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Step 2.4.4.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 2.4.4.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 2.4.4.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Step 2.4.4.2.2.2

Simplify the left side.

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Step 2.4.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.4.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Step 2.4.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 2.4.5

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

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Step 2.4.5.1

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

$x + 1 = 0$

Step 2.4.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.4.6

The final solution is all the values that make $- (3 x - 1) (x + 1) = 0$ true.

$x = \frac{1}{3}, - 1$

Step 2.5

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

$\frac{1 - 2 x}{x} = - 3 x$

Step 2.6

Find the LCD of the terms in the equation.

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Step 2.6.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 2.6.2

The LCM of one and any expression is the expression.

$x$

Step 2.7

Multiply each term in $\frac{1 - 2 x}{x} = - 3 x$ by $x$ to eliminate the fractions.

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Step 2.7.1

Multiply each term in $\frac{1 - 2 x}{x} = - 3 x$ by $x$ .

$\frac{1 - 2 x}{x} x = - 3 x \cdot x$

Step 2.7.2

Simplify the left side.

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Step 2.7.2.1

Cancel the common factor of $x$ .

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Step 2.7.2.1.1

Cancel the common factor.

$\frac{1 - 2 x}{x} x = - 3 x \cdot x$

Step 2.7.2.1.2

Rewrite the expression.

$1 - 2 x = - 3 x \cdot x$

Step 2.7.3

Simplify the right side.

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Step 2.7.3.1

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

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Step 2.7.3.1.1

Move $x$ .

$1 - 2 x = - 3 (x \cdot x)$

Step 2.7.3.1.2

Multiply $x$ by $x$ .

$1 - 2 x = - 3 x^{2}$

Step 2.8

Solve the equation.

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Step 2.8.1

Add $3 x^{2}$ to both sides of the equation.

$1 - 2 x + 3 x^{2} = 0$

Step 2.8.2

Use the quadratic formula to find the solutions.

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

Step 2.8.3

Substitute the values $a = 3$ , $b = - 2$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (3 \cdot 1)}}{2 \cdot 3}$

Step 2.8.4

Simplify.

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Step 2.8.4.1

Simplify the numerator.

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Step 2.8.4.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}$

Step 2.8.4.1.2

Multiply $- 4 \cdot 3 \cdot 1$ .

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Step 2.8.4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot 1}}{2 \cdot 3}$

Step 2.8.4.1.2.2

Multiply $- 12$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 12}}{2 \cdot 3}$

Step 2.8.4.1.3

Subtract $12$ from $4$ .

$x = \frac{2 \pm \sqrt{- 8}}{2 \cdot 3}$

Step 2.8.4.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 3}$

Step 2.8.4.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 3}$

Step 2.8.4.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{8}}{2 \cdot 3}$

Step 2.8.4.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.8.4.1.7.1

Factor $4$ out of $8$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 3}$

Step 2.8.4.1.7.2

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

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 3}$

Step 2.8.4.1.8

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{2})}{2 \cdot 3}$

Step 2.8.4.1.9

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

$x = \frac{2 \pm 2 i \sqrt{2}}{2 \cdot 3}$

Step 2.8.4.2

Multiply $2$ by $3$ .

$x = \frac{2 \pm 2 i \sqrt{2}}{6}$

Step 2.8.4.3

Simplify $\frac{2 \pm 2 i \sqrt{2}}{6}$ .

$x = \frac{1 \pm i \sqrt{2}}{3}$

Step 2.8.5

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{2}}{3}, \frac{1 - i \sqrt{2}}{3}$

Step 2.9

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

$x = \frac{1}{3}, - 1, \frac{1 + i \sqrt{2}}{3}, \frac{1 - i \sqrt{2}}{3}$