Algebra Examples

$| \frac{X}{X - 1} | = \frac{4}{X}$

Step 1

Multiply both sides by $X$ .

$| \frac{X}{X - 1} | X = \frac{4}{X} X$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Reorder factors in $| \frac{X}{X - 1} | X$ .

$X | \frac{X}{X - 1} | = \frac{4}{X} X$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $X$ .

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Step 2.2.1.1

Cancel the common factor.

$X | \frac{X}{X - 1} | = \frac{4}{X} X$

Step 2.2.1.2

Rewrite the expression.

$X | \frac{X}{X - 1} | = 4$

Step 3

Solve for $X$ .

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Step 3.1

Divide each term in $X | \frac{X}{X - 1} | = 4$ by $X$ and simplify.

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Step 3.1.1

Divide each term in $X | \frac{X}{X - 1} | = 4$ by $X$ .

$\frac{X | \frac{X}{X - 1} |}{X} = \frac{4}{X}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $X$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{X | \frac{X}{X - 1} |}{X} = \frac{4}{X}$

Step 3.1.2.1.2

Divide $| \frac{X}{X - 1} |$ by $1$ .

$| \frac{X}{X - 1} | = \frac{4}{X}$

Step 3.2

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

$\frac{X}{X - 1} = \pm \frac{4}{X}$

Step 3.3

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

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Step 3.3.1

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

$\frac{X}{X - 1} = \frac{4}{X}$

Step 3.3.2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$X \cdot X = (X - 1) \cdot 4$

Step 3.3.3

Solve the equation for $X$ .

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Step 3.3.3.1

Simplify $X \cdot X$ .

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Step 3.3.3.1.1

Rewrite.

$0 + 0 + X \cdot X = (X - 1) \cdot 4$

Step 3.3.3.1.2

Simplify by adding zeros.

$X \cdot X = (X - 1) \cdot 4$

Step 3.3.3.1.3

Multiply $X$ by $X$ .

$X^{2} = (X - 1) \cdot 4$

Step 3.3.3.2

Simplify $(X - 1) \cdot 4$ .

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Step 3.3.3.2.1

Apply the distributive property.

$X^{2} = X \cdot 4 - 1 \cdot 4$

Step 3.3.3.2.2

Simplify the expression.

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Step 3.3.3.2.2.1

Move $4$ to the left of $X$ .

$X^{2} = 4 \cdot X - 1 \cdot 4$

Step 3.3.3.2.2.2

Multiply $- 1$ by $4$ .

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

Step 3.3.3.3

Subtract $4 X$ from both sides of the equation.

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

Step 3.3.3.4

Add $4$ to both sides of the equation.

$X^{2} - 4 X + 4 = 0$

Step 3.3.3.5

Factor using the perfect square rule.

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Step 3.3.3.5.1

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

$X^{2} - 4 X + 2^{2} = 0$

Step 3.3.3.5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 X = 2 \cdot X \cdot 2$

Step 3.3.3.5.3

Rewrite the polynomial.

$X^{2} - 2 \cdot X \cdot 2 + 2^{2} = 0$

Step 3.3.3.5.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = X$ and $b = 2$ .

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

Step 3.3.3.6

Set the $X - 2$ equal to $0$ .

$X - 2 = 0$

Step 3.3.3.7

Add $2$ to both sides of the equation.

$X = 2$

Step 3.3.4

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

$\frac{X}{X - 1} = - \frac{4}{X}$

Step 3.3.5

$X \cdot X = (X - 1) \cdot - 4$

Step 3.3.6

Solve the equation for $X$ .

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Step 3.3.6.1

Simplify $X \cdot X$ .

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Step 3.3.6.1.1

Rewrite.

$0 + 0 + X \cdot X = (X - 1) \cdot - 4$

Step 3.3.6.1.2

Simplify by adding zeros.

$X \cdot X = (X - 1) \cdot - 4$

Step 3.3.6.1.3

Multiply $X$ by $X$ .

$X^{2} = (X - 1) \cdot - 4$

Step 3.3.6.2

Simplify $(X - 1) \cdot - 4$ .

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Step 3.3.6.2.1

Apply the distributive property.

$X^{2} = X \cdot - 4 - 1 \cdot - 4$

Step 3.3.6.2.2

Simplify the expression.

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Step 3.3.6.2.2.1

Move $- 4$ to the left of $X$ .

$X^{2} = - 4 \cdot X - 1 \cdot - 4$

Step 3.3.6.2.2.2

Multiply $- 1$ by $- 4$ .

$X^{2} = - 4 X + 4$

Step 3.3.6.3

Add $4 X$ to both sides of the equation.

$X^{2} + 4 X = 4$

Step 3.3.6.4

Subtract $4$ from both sides of the equation.

$X^{2} + 4 X - 4 = 0$

Step 3.3.6.5

Use the quadratic formula to find the solutions.

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

Step 3.3.6.6

Substitute the values $a = 1$ , $b = 4$ , and $c = - 4$ into the quadratic formula and solve for $X$ .

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

Step 3.3.6.7

Simplify.

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Step 3.3.6.7.1

Simplify the numerator.

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Step 3.3.6.7.1.1

Raise $4$ to the power of $2$ .

$X = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 3.3.6.7.1.2

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

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Step 3.3.6.7.1.2.1

Multiply $- 4$ by $1$ .

$X = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 4}}{2 \cdot 1}$

Step 3.3.6.7.1.2.2

Multiply $- 4$ by $- 4$ .

$X = \frac{- 4 \pm \sqrt{16 + 16}}{2 \cdot 1}$

Step 3.3.6.7.1.3

Add $16$ and $16$ .

$X = \frac{- 4 \pm \sqrt{32}}{2 \cdot 1}$

Step 3.3.6.7.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 3.3.6.7.1.4.1

Factor $16$ out of $32$ .

$X = \frac{- 4 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 3.3.6.7.1.4.2

Rewrite $16$ as $4^{2}$ .

$X = \frac{- 4 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 3.3.6.7.1.5

Pull terms out from under the radical.

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

Step 3.3.6.7.2

Multiply $2$ by $1$ .

$X = \frac{- 4 \pm 4 \sqrt{2}}{2}$

Step 3.3.6.7.3

Simplify $\frac{- 4 \pm 4 \sqrt{2}}{2}$ .

$X = - 2 \pm 2 \sqrt{2}$

Step 3.3.6.8

The final answer is the combination of both solutions.

$X = - 2 + 2 \sqrt{2}, - 2 - 2 \sqrt{2}$

Step 3.3.7

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

$X = 2, - 2 + 2 \sqrt{2}, - 2 - 2 \sqrt{2}$

Step 4

Exclude the solutions that do not make $| \frac{X}{X - 1} | = \frac{4}{X}$ true.

$X = 2, - 2 + 2 \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$X = 2, - 2 + 2 \sqrt{2}$

Decimal Form:

$X = 2, 0.82842712 \dots$