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Adding fractions might seem tricky at first, but with the right approach, it becomes quite manageable. When adding fractions, you need to add their numerators while keeping a common denominator the same. In our problem, you encounter two fractions, \( \frac{4x}{3} \) and \( \frac{x}{6} \). Notice how their denominators differ. We can't add them directly yet because of this difference.
Instead, we'll align their denominators first, which leads us to the next important concept.

Finding a common denominator is crucial when combining fractions. It's similar to finding a common language so both can be understood together. Here, our denominators are 3 and 6. The easiest way to find the least common denominator is to determine the least common multiple (LCM) of the two numbers.
In this case, the LCM of 3 and 6 is 6. Why 6? Because 6 is the smallest number that both 3 and 6 can divide into without leaving a remainder.
Once identified, we transform each fraction so they share this common denominator. For \( \frac{4x}{3} \), multiply both the numerator and denominator by 2, giving us \( \frac{8x}{6} \). The second fraction \( \frac{x}{6} \) remains unchanged because it already uses 6 as its denominator.

• Multiply fractions to reach the common denominator.
• The LCM is used to harmonize fractions.

After adding the fractions, you end up with \( \frac{9x}{6} \). This result is accurate, but it can be further simplified to make it neater. To simplify a fraction, both its numerator and denominator should be divided by their greatest common divisor (GCD).
In the case of \( 9x \) and 6, the GCD is 3. By dividing both the numerator \( 9x \) and the denominator 6 by 3, you simplify the fraction to \( \frac{3x}{2} \).

• Identify the GCD of both numerator and denominator.
• Divide to simplify.
• Result: cleaner, simpler fractions.