Problem 37 - Algebra and Trigonometry

Question

add or subtract as indicated. $$ \frac{4 x-10}{x-2}-\frac{x-4}{x-2} $$

Step-by-Step Solution

Answer

The simplified form of \(\frac{4x - 10}{x - 2} - \frac{x - 4}{x - 2}\) is \(\frac{3x - 6}{x - 2}\)

Step 1: Identify common denominators

Notice that both fractions, \(\frac{4x - 10}{x - 2}\) and \(\frac{x - 4}{x - 2}\), share the same denominator, which is \(x - 2\). This makes the addition and subtraction process more straightforward.

Step 2: Subtract fractions

As both fractions have the same denominator, the subtraction can be simply done by subtracting the numerators. Thus it becomes: \(\frac{4x - 10 - (x - 4)}{x - 2}\). Notice the inclusion of (x-4) in parentheses. This is because the negative sign in front applies to the entire expression.

Step 3: Simplify the numerator

Next, combine like terms in the numerator: \(\frac{3x - 6}{x - 2}\)

Step 4: Further Simplification

If possible, you would then simplify the fraction, but, in this case, it cannot be reduced any further, so the fraction remains as \(\frac{3x - 6}{x - 2}\)

Key Concepts

Common Denominators

The concept of common denominators is central to adding and subtracting fractions. When fractions share the same denominator, this commonality allows for direct manipulation of the numerators. For instance, when subtracting the fractions \(\frac{4 x-10}{x-2}\) and \(\frac{x-4}{x-2}\), the shared denominator \(x-2\) enables a straightforward subtraction process. It's analogous to sharing a common language; communication—or in this case, calculation—becomes much easier.

Always ensure that you have common denominators before attempting to add or subtract fractions. If the denominators differ, you would first need to find an equivalent fraction for each that has a shared common denominator. The beauty of shared denominators lies in their power to transform a potentially complex problem into a simpler one.

Simplifying Algebraic Expressions

The act of simplifying algebraic expressions is pivotal in making complex mathematical problems more manageable. Through this process, expressions are reduced to their simplest form, making it easier to understand and solve the underlying equations or inequalities. Simplifying often involves combining like terms, distributing multiples, and factoring among others.

To illustrate this with our example, \(\frac{4 x-10 - (x - 4)}{x - 2}\) simplifies to \(\frac{3x - 6}{x - 2}\) by combining like terms in the numerator. This reduces clutter in the expression and unveils the more streamlined, easily interpretable form of the fraction. Simplification is a cornerstone of algebra that aids in unraveling the knots of complexity.

Like Terms in Algebra

Understanding like terms in algebra is crucial for the simplification of algebraic expressions. Like terms are terms that contain the same variables raised to the same power. Only like terms can be combined through addition or subtraction, much like you can only add or subtract apples with apples, not with oranges.

Let's consider the numerator in our example: \(4x - 10 - (x - 4)\). Here, the terms \(4x\) and \(x\) are like terms since they both contain the variable \(x\) raised to the first power. Similarly, the constant terms \(10\) and \(4\) are like terms because they are both constants without variables. Combining these like terms yields a new expression \(3x - 6\), which is more straightforward to work with. Identifying and combining like terms is a fundamental step in the journey toward mastering algebra.