Problem 35 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \frac{4+x}{x-9}+\frac{6+x}{x-9}-\frac{1-x}{x-9} $$

Step-by-Step Solution

Answer

\(\frac{9 + 3x}{x - 9}\)

Step 1: Identify common denominators

All the fractional terms in the expression \(\frac{4+x}{x-9}+\frac{6+x}{x-9}-\frac{1-x}{x-9}\) share the same denominator which is \(x-9\). This is a straightforward fraction addition and subtraction operation because the denominators are the same.

Step 2: Addition and Subtraction of terms

Since the denominators are common, the numerators can be directly added: \((4 + x) + (6 + x) - (1 - x) = 4 + x + 6 + x - 1 + x\). This is a simple arithmetic operation.

Step 3: Simplification of expression

Add and subtract the like terms to simplify the expression. The calculation will yield \(4 + x + 6 + x - 1 + x = 9 + 3x\). So, the expression \(\frac{4+x}{x-9}+\frac{6+x}{x-9}-\frac{1-x}{x-9}\) simplifies to \(\frac{9 + 3x}{x - 9}\). This is the simplest form of the given expression.

Key Concepts

Common Denominators

Understanding common denominators is crucial when working with fractions. A denominator represents the total number of equal parts the whole is divided into. In algebra, when fractions have the same denominators, we refer to these as common denominators. For instance, consider the fractions \(\frac{4+x}{x-9}\), \(\frac{6+x}{x-9}\), and \(\frac{1-x}{x-9}\). The denominator in all of these fractions is \(x-9\), which means they have a common denominator.

Hence, when you add or subtract fractions with common denominators, you do not need to alter the denominator at all. You simply perform the addition or subtraction on the numerators (the top numbers) and keep the shared denominator as is. This process greatly simplifies the operation, making it more manageable and easier to understand.

Fraction Addition and Subtraction

When adding or subtracting fractions, the process depends on whether the denominators are the same or different. With common denominators, the process is straightforward: keep the denominator the same and add or subtract the numerators.

In our example, you have the expressions \(\frac{4+x}{x-9}\), \(\frac{6+x}{x-9}\), and \(\frac{1-x}{x-9}\). Adding or subtracting these is as simple as combining the numerators: \(4 + x + 6 + x - 1 + x\). Without common denominators, you would first need to find a common denominator before you can add or subtract, which involves more steps and can be more complex. Always look for common denominators to make the work simpler.

Like Terms

Algebra often involves combining like terms, which are terms that have the same variables raised to the same power. In our example, the like terms are the constant numbers and the terms with the variable \(x\).

To simplify an expression, combine these like terms by performing addition or subtraction as appropriate. For instance, \(4 + 6 - 1\) can be combined because they are all constants, and \(x + x + x\) because they are all linear terms of \(x\). Combining the like terms \(4 + x + 6 + x - 1 + x\) simplifies to \(9 + 3x\), revealing the algebraic expression in its simplest form. Grasping the concept of like terms helps students reduce complex expressions to simpler forms, making them easier to work with and understand.