Calculus Examples

$\frac{x + 1}{x - 1} + \frac{x - 1}{x + 1} = \frac{13}{6}$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

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

$x - 1, x + 1, 6$

Step 1.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.3

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.4

$6$ has factors of $2$ and $3$ .

$2 \cdot 3$

Step 1.5

Multiply $2$ by $3$ .

$6$

Step 1.6

The factor for $x - 1$ is $x - 1$ itself.

$(x - 1) = x - 1$

$(x - 1)$ occurs $1$ time.

Step 1.7

The factor for $x + 1$ is $x + 1$ itself.

$(x + 1) = x + 1$

$(x + 1)$ occurs $1$ time.

Step 1.8

The LCM of $x - 1, x + 1$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(x - 1) (x + 1)$

Step 1.9

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$6 (x - 1) (x + 1)$

Step 2

Multiply each term in $\frac{x + 1}{x - 1} + \frac{x - 1}{x + 1} = \frac{13}{6}$ by $6 (x - 1) (x + 1)$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{x + 1}{x - 1} + \frac{x - 1}{x + 1} = \frac{13}{6}$ by $6 (x - 1) (x + 1)$ .

$\frac{x + 1}{x - 1} (6 (x - 1) (x + 1)) + \frac{x - 1}{x + 1} (6 (x - 1) (x + 1)) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Rewrite using the commutative property of multiplication.

$6 \frac{x + 1}{x - 1} ((x - 1) (x + 1)) + \frac{x - 1}{x + 1} (6 (x - 1) (x + 1)) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.2

Combine $6$ and $\frac{x + 1}{x - 1}$ .

$\frac{6 (x + 1)}{x - 1} ((x - 1) (x + 1)) + \frac{x - 1}{x + 1} (6 (x - 1) (x + 1)) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.3

Cancel the common factor of $x - 1$ .

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Step 2.2.1.3.1

Cancel the common factor.

$\frac{6 (x + 1)}{x - 1} ((x - 1) (x + 1)) + \frac{x - 1}{x + 1} (6 (x - 1) (x + 1)) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.3.2

Rewrite the expression.

$6 (x + 1) (x + 1) + \frac{x - 1}{x + 1} (6 (x - 1) (x + 1)) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.4

Raise $x + 1$ to the power of $1$ .

$6 ({(x + 1)}^{1} (x + 1)) + \frac{x - 1}{x + 1} (6 (x - 1) (x + 1)) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.5

Raise $x + 1$ to the power of $1$ .

$6 ({(x + 1)}^{1} {(x + 1)}^{1}) + \frac{x - 1}{x + 1} (6 (x - 1) (x + 1)) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.6

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

$6 {(x + 1)}^{1 + 1} + \frac{x - 1}{x + 1} (6 (x - 1) (x + 1)) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.7

Add $1$ and $1$ .

$6 {(x + 1)}^{2} + \frac{x - 1}{x + 1} (6 (x - 1) (x + 1)) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.8

Rewrite using the commutative property of multiplication.

$6 {(x + 1)}^{2} + 6 \frac{x - 1}{x + 1} ((x - 1) (x + 1)) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.9

Combine $6$ and $\frac{x - 1}{x + 1}$ .

$6 {(x + 1)}^{2} + \frac{6 (x - 1)}{x + 1} ((x - 1) (x + 1)) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.10

Cancel the common factor of $x + 1$ .

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Step 2.2.1.10.1

Factor $x + 1$ out of $(x - 1) (x + 1)$ .

$6 {(x + 1)}^{2} + \frac{6 (x - 1)}{x + 1} ((x + 1) (x - 1)) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.10.2

Cancel the common factor.

$6 {(x + 1)}^{2} + \frac{6 (x - 1)}{x + 1} ((x + 1) (x - 1)) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.10.3

Rewrite the expression.

$6 {(x + 1)}^{2} + 6 (x - 1) (x - 1) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.11

Raise $x - 1$ to the power of $1$ .

$6 {(x + 1)}^{2} + 6 ({(x - 1)}^{1} (x - 1)) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.12

Raise $x - 1$ to the power of $1$ .

$6 {(x + 1)}^{2} + 6 ({(x - 1)}^{1} {(x - 1)}^{1}) = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.13

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

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{1 + 1} = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.2.1.14

Add $1$ and $1$ .

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{2} = \frac{13}{6} (6 (x - 1) (x + 1))$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $6$ .

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Step 2.3.1.1

Factor $6$ out of $6 (x - 1) (x + 1)$ .

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{2} = \frac{13}{6} (6 ((x - 1) (x + 1)))$

Step 2.3.1.2

Cancel the common factor.

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{2} = \frac{13}{6} (6 ((x - 1) (x + 1)))$

Step 2.3.1.3

Rewrite the expression.

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{2} = 13 ((x - 1) (x + 1))$

Step 2.3.2

Expand $(x - 1) (x + 1)$ using the FOIL Method.

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Step 2.3.2.1

Apply the distributive property.

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{2} = 13 (x (x + 1) - 1 (x + 1))$

Step 2.3.2.2

Apply the distributive property.

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{2} = 13 (x \cdot x + x \cdot 1 - 1 (x + 1))$

Step 2.3.2.3

Apply the distributive property.

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{2} = 13 (x \cdot x + x \cdot 1 - 1 x - 1 \cdot 1)$

Step 2.3.3

Simplify terms.

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Step 2.3.3.1

Combine the opposite terms in $x \cdot x + x \cdot 1 - 1 x - 1 \cdot 1$ .

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Step 2.3.3.1.1

Reorder the factors in the terms $x \cdot 1$ and $- 1 x$ .

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{2} = 13 (x \cdot x + 1 x - 1 x - 1 \cdot 1)$

Step 2.3.3.1.2

Subtract $1 x$ from $1 x$ .

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{2} = 13 (x \cdot x + 0 - 1 \cdot 1)$

Step 2.3.3.1.3

Add $x \cdot x$ and $0$ .

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{2} = 13 (x \cdot x - 1 \cdot 1)$

Step 2.3.3.2

Simplify each term.

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Step 2.3.3.2.1

Multiply $x$ by $x$ .

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{2} = 13 (x^{2} - 1 \cdot 1)$

Step 2.3.3.2.2

Multiply $- 1$ by $1$ .

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{2} = 13 (x^{2} - 1)$

Step 2.3.3.3

Simplify by multiplying through.

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Step 2.3.3.3.1

Apply the distributive property.

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{2} = 13 x^{2} + 13 \cdot - 1$

Step 2.3.3.3.2

Multiply $13$ by $- 1$ .

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{2} = 13 x^{2} - 13$

Step 3

Solve the equation.

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Subtract $13 x^{2}$ from both sides of the equation.

$6 {(x + 1)}^{2} + 6 {(x - 1)}^{2} - 13 x^{2} = - 13$

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$6 ((x + 1) (x + 1)) + 6 {(x - 1)}^{2} - 13 x^{2} = - 13$

Step 3.1.2.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 3.1.2.2.1

Apply the distributive property.

$6 (x (x + 1) + 1 (x + 1)) + 6 {(x - 1)}^{2} - 13 x^{2} = - 13$

Step 3.1.2.2.2

Apply the distributive property.

$6 (x \cdot x + x \cdot 1 + 1 (x + 1)) + 6 {(x - 1)}^{2} - 13 x^{2} = - 13$

Step 3.1.2.2.3

Apply the distributive property.

$6 (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) + 6 {(x - 1)}^{2} - 13 x^{2} = - 13$

Step 3.1.2.3

Simplify and combine like terms.

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Step 3.1.2.3.1

Simplify each term.

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Step 3.1.2.3.1.1

Multiply $x$ by $x$ .

$6 (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) + 6 {(x - 1)}^{2} - 13 x^{2} = - 13$

Step 3.1.2.3.1.2

Multiply $x$ by $1$ .

$6 (x^{2} + x + 1 x + 1 \cdot 1) + 6 {(x - 1)}^{2} - 13 x^{2} = - 13$

Step 3.1.2.3.1.3

Multiply $x$ by $1$ .

$6 (x^{2} + x + x + 1 \cdot 1) + 6 {(x - 1)}^{2} - 13 x^{2} = - 13$

Step 3.1.2.3.1.4

Multiply $1$ by $1$ .

$6 (x^{2} + x + x + 1) + 6 {(x - 1)}^{2} - 13 x^{2} = - 13$

Step 3.1.2.3.2

Add $x$ and $x$ .

$6 (x^{2} + 2 x + 1) + 6 {(x - 1)}^{2} - 13 x^{2} = - 13$

Step 3.1.2.4

Apply the distributive property.

$6 x^{2} + 6 (2 x) + 6 \cdot 1 + 6 {(x - 1)}^{2} - 13 x^{2} = - 13$

Step 3.1.2.5

Simplify.

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Step 3.1.2.5.1

Multiply $2$ by $6$ .

$6 x^{2} + 12 x + 6 \cdot 1 + 6 {(x - 1)}^{2} - 13 x^{2} = - 13$

Step 3.1.2.5.2

Multiply $6$ by $1$ .

$6 x^{2} + 12 x + 6 + 6 {(x - 1)}^{2} - 13 x^{2} = - 13$

Step 3.1.2.6

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$6 x^{2} + 12 x + 6 + 6 ((x - 1) (x - 1)) - 13 x^{2} = - 13$

Step 3.1.2.7

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 3.1.2.7.1

Apply the distributive property.

$6 x^{2} + 12 x + 6 + 6 (x (x - 1) - 1 (x - 1)) - 13 x^{2} = - 13$

Step 3.1.2.7.2

Apply the distributive property.

$6 x^{2} + 12 x + 6 + 6 (x \cdot x + x \cdot - 1 - 1 (x - 1)) - 13 x^{2} = - 13$

Step 3.1.2.7.3

Apply the distributive property.

$6 x^{2} + 12 x + 6 + 6 (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) - 13 x^{2} = - 13$

Step 3.1.2.8

Simplify and combine like terms.

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Step 3.1.2.8.1

Simplify each term.

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Step 3.1.2.8.1.1

Multiply $x$ by $x$ .

$6 x^{2} + 12 x + 6 + 6 (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) - 13 x^{2} = - 13$

Step 3.1.2.8.1.2

Move $- 1$ to the left of $x$ .

$6 x^{2} + 12 x + 6 + 6 (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) - 13 x^{2} = - 13$

Step 3.1.2.8.1.3

Rewrite $- 1 x$ as $- x$ .

$6 x^{2} + 12 x + 6 + 6 (x^{2} - x - 1 x - 1 \cdot - 1) - 13 x^{2} = - 13$

Step 3.1.2.8.1.4

Rewrite $- 1 x$ as $- x$ .

$6 x^{2} + 12 x + 6 + 6 (x^{2} - x - x - 1 \cdot - 1) - 13 x^{2} = - 13$

Step 3.1.2.8.1.5

Multiply $- 1$ by $- 1$ .

$6 x^{2} + 12 x + 6 + 6 (x^{2} - x - x + 1) - 13 x^{2} = - 13$

Step 3.1.2.8.2

Subtract $x$ from $- x$ .

$6 x^{2} + 12 x + 6 + 6 (x^{2} - 2 x + 1) - 13 x^{2} = - 13$

Step 3.1.2.9

Apply the distributive property.

$6 x^{2} + 12 x + 6 + 6 x^{2} + 6 (- 2 x) + 6 \cdot 1 - 13 x^{2} = - 13$

Step 3.1.2.10

Simplify.

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Step 3.1.2.10.1

Multiply $- 2$ by $6$ .

$6 x^{2} + 12 x + 6 + 6 x^{2} - 12 x + 6 \cdot 1 - 13 x^{2} = - 13$

Step 3.1.2.10.2

Multiply $6$ by $1$ .

$6 x^{2} + 12 x + 6 + 6 x^{2} - 12 x + 6 - 13 x^{2} = - 13$

Step 3.1.3

Combine the opposite terms in $6 x^{2} + 12 x + 6 + 6 x^{2} - 12 x + 6 - 13 x^{2}$ .

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Step 3.1.3.1

Subtract $12 x$ from $12 x$ .

$6 x^{2} + 6 + 6 x^{2} + 0 + 6 - 13 x^{2} = - 13$

Step 3.1.3.2

Add $6 x^{2} + 6 + 6 x^{2}$ and $0$ .

$6 x^{2} + 6 + 6 x^{2} + 6 - 13 x^{2} = - 13$

Step 3.1.4

Add $6 x^{2}$ and $6 x^{2}$ .

$12 x^{2} + 6 + 6 - 13 x^{2} = - 13$

Step 3.1.5

Subtract $13 x^{2}$ from $12 x^{2}$ .

$- x^{2} + 6 + 6 = - 13$

Step 3.1.6

Add $6$ and $6$ .

$- x^{2} + 12 = - 13$

Step 3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 3.2.1

Subtract $12$ from both sides of the equation.

$- x^{2} = - 13 - 12$

Step 3.2.2

Subtract $12$ from $- 13$ .

$- x^{2} = - 25$

Step 3.3

Divide each term in $- x^{2} = - 25$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- x^{2} = - 25$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 25}{- 1}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 25}{- 1}$

Step 3.3.2.2

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

$x^{2} = \frac{- 25}{- 1}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 25$ by $- 1$ .

$x^{2} = 25$

Step 3.4

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

$x = \pm \sqrt{25}$

Step 3.5

Simplify $\pm \sqrt{25}$ .

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Step 3.5.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2}}$

Step 3.5.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 5$

Step 3.6

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

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Step 3.6.1

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

$x = 5$

Step 3.6.2

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

$x = - 5$

Step 3.6.3

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

$x = 5, - 5$