Problem 60 - Algebra for College Students

Question

Explain why the solution set of the equation \(\frac{x}{3}+\frac{x}{2}=\) \(\frac{5 x}{6}\) is the entire set of real numbers.

Step-by-Step Solution

Answer

The solution set is all real numbers because the equation is true for any value of \(x\).

Step 1: Understand the Given Equation

The equation given is \(\frac{x}{3} + \frac{x}{2} = \frac{5x}{6}\). We need to determine if this equation holds true for all real numbers \(x\).

Step 2: Find a Common Denominator

To simplify the equation, we need a common denominator for the fractions on the left-hand side. The denominators are 3 and 2, so the least common denominator is 6.

Step 3: Rewrite the Fractions

Convert \(\frac{x}{3}\) to \(\frac{2x}{6}\) and \(\frac{x}{2}\) to \(\frac{3x}{6}\). Add these fractions on the left-hand side: \[\frac{2x}{6} + \frac{3x}{6} = \frac{5x}{6}\].

Step 4: Simplify Both Sides

By rewriting the left-hand side, we have \(\frac{5x}{6}\), which is exactly the expression on the right-hand side of the equation.

Step 5: Compare Both Sides of the Equation

Since the left-hand side \(\frac{5x}{6}\) equals the right-hand side \(\frac{5x}{6}\), the equation \(\frac{x}{3} + \frac{x}{2} = \frac{5x}{6}\) holds for all values of \(x\).

Step 6: Conclusion: Determine the Solution Set

The equation is true for every real number \(x\) because both sides are identically equal when simplified.

Key Concepts

Fractions

Fractions represent parts of a whole. In a fraction, the top number, known as the numerator, indicates how many parts are considered. The bottom number, the denominator, shows into how many equal parts the whole is divided. When working with equations involving fractions, like in this exercise, understanding how to manipulate them is crucial. Fractions allow us to express divisions and parts efficiently.

Fractions are especially useful in linear equations, allowing us to handle numbers that aren't whole. When solving these equations, we often need to find a common denominator to combine fractions into a single expression. By doing so, we can simplify the problem and work it out step by step.

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions with different denominators. This process involves finding a shared multiple of the denominators involved, which allows you to rewrite each fraction with the new bottom number. In our problem, the fractions have denominators of 3 and 2.

The least common denominator (LCD) for these two numbers is 6. By changing both fractions to use this LCD, we can combine them because they now share a denominator.

• Convert \(\frac{x}{3}\) to \(\frac{2x}{6}\).
• Convert \(\frac{x}{2}\) to \(\frac{3x}{6}\).

This adjustment is critical because it enables the simple addition of the numerators, leading directly to a simple and unified expression to compare to the right side of the equation.

Real Numbers

Real numbers include all the numbers on the continuous number line. These encompass numbers you use every day: integers (such as -1, 0, 1), fractions, and irrational numbers (like \(\sqrt{2}\)).

In this context, the equation being verified holds true across the entire set of real numbers. This means that no matter the number selected, whether positive, negative, or zero, the equation remains balanced and valid.
This wide applicability stems from the equation being an identity: it's always true, and no restrictions on the values of \( x \) will disrupt its balance.

Solution Set

A solution set contains all possible values that satisfy the equation. In algebra, after solving an equation, we often determine what values make the equation true. This set might include specific numbers, a range, or an infinite set.

For the equation \(\frac{x}{3} + \frac{x}{2} = \frac{5x}{6}\), we've determined through simplification and comparison that all real numbers satisfy the equation. Therefore, the solution set is the entire set of real numbers.
The equality on both sides of an identity implies any real number \(x\) can replace \(x\) in the equation and uphold equality. This characteristic is what defines the infinite solution set in this scenario.