Question

In Exercises 53-62, perform the indicated operations and simplify. \(\frac{4 x}{x-2}+\frac{x}{x-2}\)

Short answer

The simplified expression is \(\frac{5x}{x-2}\).

Step-by-step solution

Identify like terms

Both terms in our expression, \(\frac{4 x}{x-2}\) and \(\frac{x}{x-2}\), have the same denominator, \(x-2\). Due to this, they can be viewed as like terms that add together directly, similar to how \(2x\) and \(3x\) would add to \(5x\).

Perform the operation

Perform the addition of the numerators while keeping the common denominator, yielding \(\frac{4x + x}{x-2}\).

Simplify

The numerator of the fraction can be simplified by combining like terms, \(4x\) and \(x\), to get \(5x\). So the expression simplifies to \(\frac{5x}{x-2}\).

Key concepts

Understanding Adding Fractions

When adding fractions, the key idea is to combine two fractions into one. For adding, it is essential that the fractions have the same denominator. This is similar to adding numbers like 1/4 and 3/4; because they both have 4 as the denominator, you can simply add their numerators.
- In the given exercise, both fractions \( \frac{4x}{x-2} \) and \( \frac{x}{x-2} \) already have the same denominator \( x-2 \).
- When denominators are the same, you add the numerators just like regular numbers. For example, if you have \( \frac{a}{b} \) and \( \frac{c}{b} \), you'd combine them to get \( \frac{a+c}{b} \). This is what happens in the exercise, where the numerators \( 4x + x \) are added to get \( 5x \).

Exploring Common Denominator

The common denominator is a crucial concept when working with fractions. Think of it as a common ground that allows you to combine two fractions seamlessly. For addition and subtraction, the fractions must share this same base number, which is known as the denominator.
- Having a common denominator simplifies the process because it lets you focus on just adding or subtracting the numerators. \( \frac{4x}{x-2} \) and \( \frac{x}{x-2} \) both share the denominator \( x-2 \), allowing for direct addition.
- If the denominators weren’t the same, you'd have to manipulate the fractions to make them equivalent, which involves finding a least common denominator. For example, converting \(\frac{2}{3}\) and \(\frac{1}{2}\) to have 6 as the common denominator leads to \(\frac{4}{6}\) and \(\frac{3}{6}\). Look at how much simpler it becomes once they’re made compatible!

Mastering Simplifying Expressions

Simplifying expressions is all about making an expression as straightforward as possible without changing its value. This often involves combining like terms and making operations result in the simplest form.
- In the exercise, after recognizing the like terms in the numerator \( 4x \) and \( x \), it's simplified to \( 5x \). This cutting down helps ensure that the expression is easy to read and understand.
- Simplifying also assists in reducing calculation mistakes later on. If you end up with a simple expression, you're less likely to make errors when performing further operations.
- Consider the simplified result \( \frac{5x}{x-2} \), a representation that clearly shows the combined input of the original fractions. Always look to condense expressions when possible, as it provides clarity and reduces confusion in problem-solving.