Problem 65 - College Algebra

Question

The equations in Exercises \(59-70\) combine the types of equations we have discussed in this section. Solve each equation or state that it is true for all real numbers or no real numbers. $$ \frac{2}{x}+\frac{1}{2}=\frac{3}{4} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x=8\).

Step 1: Identify the denominators

Looking at the equation, we have fractions with the denominators: \(x\), 2, and 4. The least common multiple of these denominators is 4x.

Step 2: Multiply every term by the least common multiple

To remove the fractions, you multiply every term by 4x to get: \(8 + 2x = 3x\).

Step 3: Rearrange to isolate x

Subtract 2x from both sides of the equation to get: \(8 = x\) which can be written simply as \(x=8\).

Key Concepts

Solving Algebraic Fractions

Algebraic fractions are fractions that contain variables in their numerators, denominators, or both. Solving equations with algebraic fractions often requires you to find a common denominator and eliminate the fractions so that you can work with a simpler, equivalent algebraic expression.

For example, consider the equation \(\frac{2}{x}+\frac{1}{2}=\frac{3}{4}\). To solve for \(x\), you would first identify and combine like terms or expand expressions to make the equation easier to work with. After finding a common denominator, you can multiply every term by that denominator to remove the fractions, which simplifies the equation and allows you to use algebraic methods to isolate the variable.

Least Common Multiple

The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the numbers. It is a crucial concept in solving equations with fractions since finding the LCM of the denominators allows you to combine them into a single denominator without changing the equation's value.

In the exercise \(\frac{2}{x}+\frac{1}{2}=\frac{3}{4}\), the denominators are \(x\), 2, and 4. The LCM of these denominators is \(4x\), which, when used to multiply all terms of the equation, clears the fractions and sets the stage for further manipulation to isolate the variable. Understanding how to find the LCM will make solving equations with multiple fractions much more manageable.

Isolating Variables in Algebra

Isolating the variable in an algebraic equation is the process of rearranging the equation to express the target variable on one side and all other terms on the opposite side. This involves moving terms across the 'equals' sign via addition, subtraction, multiplication, or division while keeping the equation balanced.

After multiplying the terms in our example by the LCM, the equation becomes \(8 + 2x = 3x\). To isolate \(x\), subtract \(2x\) from both sides to get \(8 = x\). This technique allows you to find the specific value or range of values for the variable, which is the ultimate goal in solving an algebraic equation. Mastering this skill is a foundation for success in algebra and beyond.