Problem 79 - Algebra for College Students

Question

Solve each equation. Identify each equation as a conditional equation, an inconsistent equation, or an identity. State the solution sets to the identities using interval notation. $$\frac{1}{x}+\frac{1}{2 x}=\frac{x+2}{2 x}$$

Step-by-Step Solution

Answer

The solution is \( x = 1 \), and it’s a conditional equation with the solution set \( \{ 1 \} \).

Step 1 - Combine the Fractions

Combine the terms on the left side of the equation by finding a common denominator. The common denominator for \( \frac{1}{x} \) and \( \frac{1}{2x} \) is \( 2x \). Thus, \[ \frac{2}{2x} + \frac{1}{2x} = \frac{x+2}{2x} \]

Step 2 - Simplify the Left Side

Combine the fractions on the left side: \[ \frac{2+1}{2x} = \frac{x+2}{2x} \] which simplifies to \[ \frac{3}{2x} = \frac{x+2}{2x} \]

Step 3 - Compare the Numerators

Since the denominators are the same, set the numerators equal to each other and solve for \( x \): \[ 3 = x + 2 \]

Step 4 - Solve for \( x \)

Subtract 2 from both sides of the equation: \[ 3 - 2 = x + 2 - 2 \] Thus, \[ x = 1 \]

Step 5 - Identify the Equation Type and Solution Set

Since the equation has a single solution, it is a conditional equation. The solution set is simply \( \{ 1 \} \).

Key Concepts

Conditional Equation

A conditional equation is a type of equation that is true only for certain values of the variable involved. In this exercise, the equation \(\frac{1}{x}+\frac{1}{2x}=\frac{x+2}{2x}\) is considered a conditional equation because it has a specific solution. To determine if an equation is conditional, solve it and check if the solution is valid. If the equation holds true for only one or a specific set of values, it is conditional. In this case, our solution is \(x = 1\), making it conditional. Remember: if you find no solution or the equation holds true for all values, you are dealing with an inconsistent equation or an identity respectively.

Common Denominator

When dealing with rational equations, finding a common denominator is crucial. The common denominator allows you to combine fractions by rewriting them with the same base. For example, in our original equation, we have \( \frac{1}{x} + \frac{1}{2x} \). The denominators here are \(x\) and \(2x\). The common denominator is the least common multiple (LCM) of these denominators. In this case, the LCM of \(x\) and \(2x\) is \(2x\). Once found, rewrite each fraction with the common denominator:
\(\frac{2}{2x} + \frac{1}{2x} \). This step is essential for combining the fractions accurately and simplifying the equation further.

Solution Set

The solution set of an equation consists of all the values of the variable that satisfy the equation. After solving the rational equation \(\frac{1}{x}+\frac{1}{2x}=\frac{x+2}{2x}\), we found that \(x = 1\). This means plugging \(x = 1\) back into the equation makes both sides equal, confirming it as a valid solution. Therefore, the solution set of this particular equation is \(\{1\}\). It's important to state the solution set clearly and use appropriate notation. For instance, if the solution had been all values except certain ones, we’d use interval notation to represent all possible solutions.