Problem 1 - Introductory Algebra for College Students

Question

Solve each rational equation. $$\frac{x}{3}=\frac{x}{2}-2$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 12\).

Step 1: Multiply by Common Denominator

The first step is to eliminate the fractions in the equation. This can be achieved by multiplying each term by the common denominator, which is 6 in this case. So,\[6 \cdot \frac{x}{3}=6 \cdot (\frac{x}{2}-2)\]This simplifies to \(2x = 3x - 12\).

Step 2: Rearrange the Equation

Rearrange the equation to move all the terms to one side of the equation by subtracting \(3x\) from both sides, this results in: \(-x = -12\)

Step 3: Solve for the Unknown, x

The last step is to divide the equation by \(-1\) in order to solve for \(x\). This gives \(x = 12\).

Key Concepts

Common Denominator

Rational equations often feature fractions, making them tricky to solve at first. To simplify things, we look for a common denominator. This helps us standardize each fraction, allowing us to combine or eliminate them. In the equation \(\frac{x}{3} = \frac{x}{2} - 2\), we need a common denominator to clear the fractions. The denominators are 3 and 2, and a simple way to find a common denominator is to take their least common multiple (LCM). The LCM of 3 and 2 is 6.Multiplying every term by 6 ensures all terms are in whole numbers, as it's the smallest number both 3 and 2 can divide. This eliminates the fractions and transforms the equation into a simpler form. Understanding how to find and use a common denominator is crucial because it assists in both simplifying the problem and paving the way for easier calculations.

Equation Rearrangement

Once we have multiplied to clear the fractions, the next challenge is rearranging the equation. From step 1, our equation is now \(2x = 3x - 12\). The goal here is to get all terms involving variables on one side and constants on the other.Subtract \(3x\) from both sides to start simplifying the equation:

• This gives us \(-x = -12\).

By doing this, we isolate the variable, effectively streamlining the equation and bringing us a step closer to the solution. Rearranging the equation requires practice to understand effectively which terms to move and how. It's a critical skill because it ensures consistency and accuracy as we solve for the unknown variable.

Fraction Elimination

With the fractions eliminated, we continue solving our equation. The equation \(-x = -12\) needs one final step. Here, the variable is already isolated, but it's negative. Hence, divide both sides by \(-1\) to eliminate the negative sign:

• This results in \(x = 12\).

This step, while simple, is crucial in reaching the final answer. Fraction elimination involves both clearing fractions initially and resolving any resultant negative signs. The process makes problems more straightforward to work through, especially in equations where variables hide within fraction terms. Solving rational equations thus involves clearing fractions early, ensuring the remaining steps are straightforward.