Question

Perform the indicated operations and simplify.\(\frac{3 x-2}{x+1}+\frac{2-x}{x+1}\)

Short answer

The simplified form of the expression is \( \frac{2x}{x+1}\)

Step-by-step solution

Combine Like Terms

Since both fractions have the same denominator \((x+1)\), they can directly be added together. The numerator will be the sum of the numerators from both fractions. So, \( \frac{3x-2}{x+1} + \frac{2-x}{x+1} = \frac{3x-2+2-x}{x+1}\).

Simplify the Numerator

Within the numerator, combine like terms. It gives \( \frac{2x}{x+1}\).

Simplify the Fraction

The numerator and denominator do not have any common factors apart from 1. So, the fraction \( \frac{2x}{x+1}\) is already in its simplest form.

Key concepts

Combining Like Terms

Mastering the art of combining like terms is crucial in algebra, especially when simplifying expressions and solving equations. Essentially, like terms are terms in an expression that have the same variable raised to the same power. In the context of the exercise
\(\frac{3x-2}{x+1} + \frac{2-x}{x+1}\),
we are dealing with a situation where we have terms in the numerators that can be simplified by combining them. After adding the fractions with a common denominator, we get a single fraction whose numerator is
\(3x - 2 + 2 - x\).
Here, \(3x\) and \( - x\) are like terms since they both contain the variable \(x\) raised to the first power. Combining them, we get
\(2x\),
and the constant terms \( - 2\) and \(2\) also cancel each other out. This step is essential for simplifying algebraic expressions and making them more manageable for further operations.

Simplifying the Numerator

The numerator of a fraction often needs to be simplified to reach the final answer in its simplest form. In our exercise, once the like terms have been combined, we focus on simplifying the numerator of the resulting fraction.
After combining like terms, we are left with the simplified numerator \(2x\).
This step not only makes the fraction look cleaner, but it also sets the stage for identifying any further simplifications that might be possible, such as reducing the fraction to lowest terms. Simplifying the numerator can involve combining like terms, factoring, expanding, or cancelling out terms, depending on the complexity of the expression involved.

Common Factors in Algebra

Identifying common factors is a fundamental aspect of simplifying algebraic fractions. A common factor is a number or algebraic expression that divides exactly into two or more numbers or expressions.
In the exercise, after simplifying, we arrive at the fraction \(\frac{2x}{x+1}\).
Common factors would allow us to reduce the fraction further if they existed in both the numerator and the denominator. In our case, there are no common factors besides 1, which is the omnipresent trivial factor, so the fraction cannot be reduced. Recognizing when a fraction is already simplified is just as important as being able to factor and cancel terms. In algebra, always consider if factors can be cancelled out to simplify expressions to their lowest terms, which streamlines equations and can make solving them much easier.