Question

Solve each equation, if possible. $$ \frac{1}{2}+\frac{2}{x}=\frac{3}{4} $$

Short answer

x = 8

Step-by-step solution

Isolate the fraction involving x

To start, subtract \(\frac{1}{2}\) from both sides of the equation: \(\frac{2}{x} = \frac{3}{4} - \frac{1}{2}\).

Find a common denominator

Rewrite \(\frac{3}{4} - \frac{1}{2}\) by finding a common denominator, which is 4. Thus, \(\frac{1}{2} = \frac{2}{4}\), and the equation becomes \(\frac{2}{x} = \frac{3}{4} - \frac{2}{4}\).

Simplify the right side

Subtract the fractions on the right-hand side: \(\frac{3}{4} - \frac{2}{4} = \frac{1}{4}\). Now the equation is \(\frac{2}{x} = \frac{1}{4}\).

Cross-multiply to solve for x

Cross-multiply to get rid of the fractions: \(\frac{2}{x} = \frac{1}{4}\) becomes \(\frac{2 \times 4}{1} = x\).

Simplify the result

Simplify the left side to find the value of x: \(2 \times 4 = 8\). Therefore, \(x = 8\).

Key concepts

Fraction Operations

Understanding how to handle fractions is essential in solving equations involving them. Fractions are simply divisions of integers. For instance, \(\frac{1}{2}\) means 1 divided by 2. In equations, fractions can add, subtract, multiply, or divide. Here are some important points:

• When adding or subtracting fractions, we need a common denominator.
• When multiplying fractions, multiply the numerators together and the denominators together.
• When dividing fractions, multiply by the reciprocal of the divisor.

In the given exercise, we first subtract \(\frac{1}{2}\) from both sides to isolate the term involving x, setting up for further fraction operations.

Cross-Multiplication

Cross-multiplication is a method used to solve equations where fractions are set equal to each other. It helps eliminate fractions, making the equation easier to solve. For the equation \(\frac{2}{x} = \frac{1}{4}\):

• Multiply the numerator of the first fraction by the denominator of the second (2 \cdot\ 4).
• Then, multiply the denominator of the first fraction by the numerator of the second (x \cdot\ 1).

The result of cross-multiplying \(\frac{2}{x} = \frac{1}{4}\) leads to \[2 \cdot 4 = x \cdot 1\rightarrow 8 = x\rightarrow x = 8.\] This method is particularly effective in equations where fractions are compared.

Isolating Variables

Isolating variables means to get the variable (often x or y) alone on one side of the equation. This process simplifies the equation, allowing us to solve for the unknown variable. In our exercise:

• We started by subtracting \(\frac{1}{2}\) from both sides to isolate the fraction involving x: \(\frac{2}{x} = \frac{3}{4} - \frac{1}{2}\).
• After that, simplify the right side and use cross-multiplication.

Every move we make aims at isolating x to find its value. Breaking down the problem into smaller steps is key.

Common Denominators

A common denominator is necessary to perform addition or subtraction of fractions. It’s the same denominator shared between different fractions. Here’s what happens in our example:

• Our fractions are \(\frac{3}{4}\) and \(\frac{1}{2}\), which have 4 as the least common denominator.
• We rewrite \(\frac{1}{2}\) as \(\frac{2}{4}\) to prepare for subtraction.

So, \(\frac{3}{4} - \frac{2}{4} = \frac{1}{4}\). Using common denominators helps simplifying fraction operations and solving the equation.