Problem 9 - Introductory Algebra for College Students

Question

Solve each rational equation. $$\frac{2}{x}+3=\frac{5}{2 x}+\frac{13}{4}$$

Step-by-Step Solution

Answer

The solution to the given rational equation is \( x = -2 \).

Step 1: Find a common denominator

In the given equation \( \frac{2}{x} + 3 = \frac{5}{2x} + \frac{13}{4} \), the denominators are \( x \), \( 2x \), and 4. The least common multiple of all denominators is \( 4x \).

Step 2: Simplify the equation

Multiply every term by adopted least common multiple \( 4x \), which give: \( 8 + 12x =10+13x \).

Step 3: Bring like terms together and solve

Combine like terms on both sides of the equation, which yields: \( x = -2 \).

Key Concepts

Least Common Multiple

When working with rational equations, especially those involving fractions, the least common multiple (LCM) plays a crucial role. The LCM is the smallest number that is evenly divisible by all denominators in the equation. Finding the LCM allows us to eliminate the fractions and simplify solving the equation.

In the example given, we have the denominators \( x \), \( 2x \), and 4. To find the LCM:

• Identify the highest power of each variable and number appearing in the denominators.
• Here, \( x \) is the variable with the highest power, and 4 is the largest number.
• The LCM becomes \( 4x \).

By determining the LCM, you form a common ground for all terms, making the next step, multiplying each term by the LCM, possible. This results in a much simpler equation without fractions.

Solving Equations

Solving equations is about finding the unknown variable value that satisfies the equation. Once we have rewritten the equation without fractions, solving becomes more straightforward.

In our equation, after using the LCM (\(4x\)), the rational equation \( \frac{2}{x} + 3 = \frac{5}{2x} + \frac{13}{4} \) becomes \( 8 + 12x = 10 + 13x \). At this stage:

• Start isolating the variable, which is usually represented as \( x \).
• To solve, rearrange the terms so that all \( x \)-terms are on one side of the equation and constants on the other.

This crucial manipulation allows for an easier solution. As seen in the example, we finish with \( x = -2 \). Solving these steps clearly highlights how combining terms aids in reaching a solution.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate the variable you are solving for. Skills like adding, subtracting, multiplying, or dividing terms on both sides of the equation help simplify it.

In our rational equation, after transforming it post-LCM multiplication, we have:

• First, connect like terms on both sides to simplify. This means aligning terms with \( x \) and constants separately.
• Subtract \( 12x \) from both sides: \( 8 = 10 + x \).
• Subtracting 10 from both sides simplifies to \( -2 = x \).

Every step follows simple operations that maintain the equation's balance. Algebraic manipulation is a fundamental skill in mathematics, needed for solving complex equations and creating solutions efficiently.