Problem 15 - Introductory Algebra for College Students

Question

Solve each rational equation. $$\frac{x-2}{2 x}+1=\frac{x+1}{x}$$

Step-by-Step Solution

Answer

The solution for x is 4.

Step 1: Write down the equation

The initial equation is \(\frac{x-2}{2x}+1=\frac{x+1}{x}\).

Step 2: Find the Least Common Multiple (LCM)

In this case, the denominators are 2x and x. Their least common multiple (LCM) would be 2x.

Step 3: Multiply each term by the LCM

The equation becomes \(2x*\frac{x-2}{2x}+2x*1=2x*\frac{x+1}{x}\). This simplifies to \(x-2+2x=2(x+1)\).

Step 4: Simplify the equation

The equation simplifies to \(3x-2=2x+2\).

Step 5: Solve for x

By further simplifying, x equals to 4.

Key Concepts

Least Common Multiple (LCM)

The Least Common Multiple (LCM) is a concept often used in solving rational equations. It helps us eliminate fractions, making equations easier to handle.
When given fractions with different denominators, the LCM is the smallest number that both denominators can divide into without leaving a remainder.

• To find the LCM of two or more numbers, list their multiples first.
• Locate the smallest multiple that appears in all lists.

For the equation \(\frac{x-2}{2x}+1=\frac{x+1}{x}\), the denominators are \(2x\) and \(x\).

• The LCM of \(2x\) and \(x\) is \(2x\).

By using the LCM, you can create a uniform denominator, allowing for easier manipulation of the equation.

Simplifying Equations

Simplifying equations is a key step that transforms complex mathematical expressions into simpler ones. After dealing with the LCM, the equation becomes cleaner and more manageable.
Let's break down the simplification process:

• Multiply each term by the LCM to eliminate denominators.
• Combine like terms where possible.

For example, multiplying every term in \(\frac{x-2}{2x}+1=\frac{x+1}{x}\) by the LCM \(2x\) gives us:

• \(x-2+2x=2(x+1)\).

Now you can easily combine like terms, such as \(x\) and \(2x\), resulting in \(3x-2=2x+2\). This further simplifies the equation without affecting its solutions.

Solving for Variables

Once the equation is simplified, the final step is solving for the unknown variable, often represented as \(x\).
In the simplified equation \(3x-2=2x+2\), the goal is to isolate \(x\) on one side of the equation.

• First, subtract \(2x\) from both sides to focus on \(x\).
• You get \(x-2=2\).
• Then, add 2 to both sides to solve for \(x\), resulting in \(x=4\).

This straightforward approach lets you find the value of the variable, which is the solution to the rational equation. Practice helps in understanding and becoming quicker at recognizing the proper steps to isolate and solve for the required variable.