Problem 67 - Introductory Algebra for College Students

Question

Describe similarities and differences between the procedures needed to solve the following problems: $$ \text { Add: } \frac{2}{x}+\frac{3}{4}, \quad \text { Solve: } \frac{2}{x}+\frac{3}{4}=1 $$

Step-by-Step Solution

Answer

For the addition, result is \(\frac{2+3x}{4x}\), and for the equation, the solution is \(x = 2\). The procedures are similar in the common denominator aspect, but differentiate when the second problem requires solving for 'x'.

Step 1: Addition of fractions

The first problem asks to add two fractions. The standard approach to add fractions is to procure a common denominator, which usually is the product of the denominators. So, The result is \(\frac{2}{x} + \frac{3x}{4x} = \frac{2+3x}{4x}\), assuming \(x \neq 0\).

Step 2: Solving the Equation

The second problem is an equation and requires solving for 'x'. Having already calculated the sum as \(\frac{2+3x}{4x}\) in Step 1, equate this to 1: \(\frac{2+3x}{4x} = 1\). Multiply the entire equation by \(4x\) to get rid of the denominator on the left side leading to \(2+3x = 4x\). Once we have this, we can isolate 'x' by subtracting \(3x\) from both sides of the equation resulting in \(2 = x\).

Step 3: Comparison

When comparing these two procedures, the initial step is the same in both cases: finding a common denominator to add the fractions. However, the procedures diverge when the equation in the second problem must be solved for 'x', adding the steps of equating the expression to 1, removing the denominator on one side, and isolating 'x'.