Problem 14 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{3 x}{2 x+1}-\frac{5}{2 x+1} $$

Step-by-Step Solution

Answer

\( \frac{3x - 5}{2x + 1} \)

Step 1: Identify the Common Denominator

Notice that both fractions share the same denominator, which is \(2x + 1\). This means that we do not need to find a new common denominator for these expressions.

Step 2: Combine the Numerators

Since the denominators are the same, we can directly combine the numerators. This gives us:\[\frac{3x - 5}{2x + 1}\]

Step 3: Simplify the Expression (if possible)

Check if the numerator \(3x - 5\) can be factored or simplified further, and see if there are common factors between the numerator and the denominator. In this case, \(3x - 5\) does not factor in a way that allows for cancellation with the denominator. Thus, the expression is already in its simplest form.

Key Concepts

Adding Rational Expressions

Adding rational expressions is a lot like adding regular fractions. The key is to have a common denominator. Once you have it, you can simply add the numerators together.
When the denominators are already the same, your job is much easier. You just need to add the numerators and keep the common denominator the same.
For example, if you have \(\frac{a}{b} + \frac{c}{b}\), you can write this as \(\frac{a+c}{b}\).
Things get a bit trickier when the denominators differ. You'll have to find the least common denominator, convert each rational expression, and then add their numerators.

Subtracting Rational Expressions

Subtracting rational expressions follows rules similar to adding them, but with that extra twist: taking a difference requires careful attention.
Just like with addition, you need a common denominator. When you have the same denominator, you simply subtract the numerators.
For example, with \(\frac{a}{b} - \frac{c}{b}\), the result is \(\frac{a-c}{b}\).
In our exercise, the rational expressions, \(\frac{3x}{2x+1}-\frac{5}{2x+1}\\), already have the same denominator. That makes it easy! You subtract the numerators: \(3x - 5\).
Again, if the denominators differ, it involves finding the least common denominator first, transforming each expression, and then subtracting the numerators.

Simplifying Rational Expressions

Simplifying rational expressions is an important step to make sure the answer is neat and concise.
The aim during simplification is to make the expression as simple as possible by reducing it. You do this by finding common factors in the numerator and the denominator.
Keep in mind, sometimes you can factor both the numerator and the denominator. This allows you to cancel out any common factors, making the expression simpler.
In the exercise given, we reached \(\frac{3x - 5}{2x + 1}\). You check if \(3x - 5\) can be factored to cancel with something in the denominator. But, since it doesn't factor further for cancellation with \(2x + 1\), the expression is already simplified. Remember, the goal of simplification is to make sure no further factors can be cancelled.