Solve by Factoring 2/x+1/2=5/x

$\frac{2}{x} + \frac{1}{2} = \frac{5}{x}$

Move $\frac{5}{x}$ to the left side of the equation by subtracting it from both sides.

$\frac{2}{x} + \frac{1}{2} - \frac{5}{x} = 0$

Simplify $\frac{2}{x} + \frac{1}{2} - \frac{5}{x}$ .

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Combine fractions with similar denominators.

$\frac{2 - 5}{x} + \frac{1}{2} = 0$

Simplify the expression.

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Subtract $5$ from $2$ .

$\frac{- 3}{x} + \frac{1}{2} = 0$

Move the negative in front of the fraction.

$- \frac{3}{x} + \frac{1}{2} = 0$

Combine fractions with similar denominators.

$\frac{2 - 5}{x} + \frac{1}{2} = 0$

Simplify the expression.

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Subtract $5$ from $2$ .

$\frac{- 3}{x} + \frac{1}{2} = 0$

Move the negative in front of the fraction.

$- \frac{3}{x} + \frac{1}{2} = 0$

To write $- \frac{3}{x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3}{x} \cdot \frac{2}{2} + \frac{1}{2} = 0$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{3}{x} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{x}{x} = 0$

Write each expression with a common denominator of $x \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{3}{x}$ and $\frac{2}{2}$ .

$- \frac{3 \cdot 2}{x \cdot 2} + \frac{1}{2} \cdot \frac{x}{x} = 0$

Multiply $\frac{1}{2}$ and $\frac{x}{x}$ .

$- \frac{3 \cdot 2}{x \cdot 2} + \frac{x}{2 x} = 0$

Reorder the factors of $x \cdot 2$ .

$- \frac{3 \cdot 2}{2 x} + \frac{x}{2 x} = 0$

Combine the numerators over the common denominator.

$\frac{- 3 \cdot 2 + x}{2 x} = 0$

Multiply $- 3$ by $2$ .

$\frac{- 6 + x}{2 x} = 0$

Find the LCD of the terms in the equation.

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

$2 x, 1$

Since $2 x, 1$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $2, 1$ then find LCM for the variable part $x^{1}$ .

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.

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

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

Not prime

The LCM of $2, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

The LCM of $x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x$

The LCM for $2 x, 1$ is the numeric part $2$ multiplied by the variable part.

$2 x$

Multiply each term by $2 x$ and simplify.

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Multiply each term in $\frac{- 6 + x}{2 x} = 0$ by $2 x$ in order to remove all the denominators from the equation.

$\frac{- 6 + x}{2 x} \cdot (2 x) = 0 \cdot (2 x)$

Simplify $\frac{- 6 + x}{2 x} \cdot (2 x)$ .

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Rewrite using the commutative property of multiplication.

$2 \frac{- 6 + x}{2 x} x = 0 \cdot (2 x)$

Cancel the common factor of $2$ .

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Factor $2$ out of $2 x$ .

$2 \frac{- 6 + x}{2 (x)} x = 0 \cdot (2 x)$

Cancel the common factor.

$2 \frac{- 6 + x}{2 x} x = 0 \cdot (2 x)$

Rewrite the expression.

$\frac{- 6 + x}{x} x = 0 \cdot (2 x)$

Cancel the common factor of $x$ .

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Cancel the common factor.

$\frac{- 6 + x}{x} x = 0 \cdot (2 x)$

Rewrite the expression.

$- 6 + x = 0 \cdot (2 x)$

Multiply $0 (2 x)$ .

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Multiply $2$ by $0$ .

$- 6 + x = 0 x$

Multiply $0$ by $x$ .

$- 6 + x = 0$

Add $6$ to both sides of the equation.

$x = 6$

Solve by Factoring 2/x+1/2=5/x

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