91Ó°ÊÓ

When adding or subtracting rational expressions, the first step is to find a common denominator. It's similar to what you do with regular fractions. A common denominator means both fractions have the same bottom part, or denominator. In our exercise, both fractions \(\frac{5x+4}{6x+5}\) and \(\frac{x+1}{6x+5}\) already have the same denominator, which is \(6x+5\). So, we're all set to move on to the next step.

If they didn't have the same denominator, we would need to find the least common denominator (LCD). This might involve factoring the denominators and finding a common multiple. But in this case, it's straightforward because the denominators are already the same. This makes our job easier.

Once we have a common denominator, our focus shifts to combining the numerators. The numerators are simply the top parts of the fractions. Since the denominators are the same, we can directly add the numerators together.

Here's the step for our exercise: we combine the numerators of \(\frac{5x+4}{6x+5}\) and \(\frac{x+1}{6x+5}\). The combined numerator becomes: \((5x + 4) + (x + 1)\).

Remember, to correctly combine them, we add the like terms: \((5x + x)\) and \((4 + 1)\). This simplification inside the numerator gives us \(6x + 5\). So, we rewrite our expression as: \(\frac{6x + 5}{6x+5}\).

The final step is to simplify the fraction. We've combined the numerators and have \(\frac{6x + 5}{6x + 5}\). Here, the numerator and the denominator are the same. When a fraction's numerator and denominator are identical, it simplifies to 1. So, \(\frac{6x + 5}{6x + 5} = 1\).

This simplification step is crucial as it checks our final result. There's no need for further reduction since we've reached the simplest form.

In summary, when dealing with adding and subtracting rational expressions:

• First, find or confirm the common denominator.
• Next, combine the numerators.
• Lastly, simplify the fraction.

Mastering these steps will make your work with rational expressions much smoother.