Algebra Examples

$\frac{8}{4 u - 8} - 1 = \frac{2}{u - 2}$

Step 1

Add $1$ to both sides of the equation.

$\frac{8}{4 u - 8} = \frac{2}{u - 2} + 1$

Step 2

Factor each term.

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Step 2.1

Factor $4$ out of $4 u - 8$ .

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Step 2.1.1

Factor $4$ out of $4 u$ .

$\frac{8}{4 (u) - 8} = \frac{2}{u - 2} + 1$

Step 2.1.2

Factor $4$ out of $- 8$ .

$\frac{8}{4 u + 4 \cdot - 2} = \frac{2}{u - 2} + 1$

Step 2.1.3

Factor $4$ out of $4 u + 4 \cdot - 2$ .

$\frac{8}{4 (u - 2)} = \frac{2}{u - 2} + 1$

Step 2.2

Reduce the expression $\frac{8}{4 (u - 2)}$ by cancelling the common factors.

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Step 2.2.1

Factor $4$ out of $8$ .

$\frac{4 \cdot 2}{4 (u - 2)} = \frac{2}{u - 2} + 1$

Step 2.2.2

Cancel the common factor.

$\frac{4 \cdot 2}{4 (u - 2)} = \frac{2}{u - 2} + 1$

Step 2.2.3

Rewrite the expression.

$\frac{2}{u - 2} = \frac{2}{u - 2} + 1$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

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

$u - 2, u - 2, 1$

Step 3.2

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 3.3

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

Not prime

Step 3.4

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

$1$

Step 3.5

The factor for $u - 2$ is $u - 2$ itself.

$(u - 2) = u - 2$

$(u - 2)$ occurs $1$ time.

Step 3.6

The LCM of $u - 2, u - 2$ is the result of multiplying all factors the greatest number of times they occur in either term.

$u - 2$

Step 4

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

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Step 4.1

Multiply each term in $\frac{2}{u - 2} = \frac{2}{u - 2} + 1$ by $u - 2$ .

$\frac{2}{u - 2} (u - 2) = \frac{2}{u - 2} (u - 2) + 1 (u - 2)$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $u - 2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2}{u - 2} (u - 2) = \frac{2}{u - 2} (u - 2) + 1 (u - 2)$

Step 4.2.1.2

Rewrite the expression.

$2 = \frac{2}{u - 2} (u - 2) + 1 (u - 2)$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Cancel the common factor of $u - 2$ .

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Step 4.3.1.1.1

Cancel the common factor.

$2 = \frac{2}{u - 2} (u - 2) + 1 (u - 2)$

Step 4.3.1.1.2

Rewrite the expression.

$2 = 2 + 1 (u - 2)$

Step 4.3.1.2

Multiply $u - 2$ by $1$ .

$2 = 2 + u - 2$

Step 4.3.2

Combine the opposite terms in $2 + u - 2$ .

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Step 4.3.2.1

Subtract $2$ from $2$ .

$2 = u + 0$

Step 4.3.2.2

Add $u$ and $0$ .

$2 = u$

Step 5

Rewrite the equation as $u = 2$ .

$u = 2$

Step 6

Exclude the solutions that do not make $\frac{8}{4 u - 8} - 1 = \frac{2}{u - 2}$ true.

$No solution$