Problem 93 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and simplify. $$12-\frac{4}{x+2}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{12x+20}{x+2}\).

Step 1: Identify the operation

The given expression is \(12-\frac{4}{x+2}\). The operation to be performed is subtraction.

Step 2: Perform the operation

To perform subtraction with fractions, you would need to have the same denominator. In this case, the number 12 can be viewed as a fraction with a denominator of 1, that is \(12 = \frac{12}{1}\). Now the denominators are different (1 and \(x+2\)). To subtract fractions, the denominators must be the same. You can rewrite 12 as \(\frac{12(x+2)}{(x+2)}\). This would result into \(\frac{12x+24}{x+2} - \frac{4}{x+2}\).

Step 3: Simplify the expression

Now that both terms have the same denominator, you can combine the numerators, resulting in \(\frac{12x+24-4}{x+2}\) which simplifies to \(\frac{12x+20}{x+2}\).

Key Concepts

Subtracting Fractions

Subtracting fractions can sometimes be perplexing, but with the right approach, anyone can master it. Imagine you're sharing a pizza with friends, and you're figuring out how much is left after everyone takes a slice. This is similar to subtracting fractions, where you're finding the difference between parts of a whole.

Firstly, ensure the fractions have the same denominator, which is like making sure each pizza slice is the same size before you subtract one from another. If the denominators don't match, find a common size for all the slices (the least common denominator). Once the fractions have equivalent denominators, subtract the numerators, which represent the actual number of slices.

For example, if you have \( \frac{3}{4} \) of your pizza left and your friend takes \( \frac{1}{2} \) of a pizza, first convert the \( \frac{1}{2} \) to match the denominator of 4, resulting in \( \frac{2}{4} \). Then subtract the numerators, giving \( \frac{3}{4} - \frac{2}{4} = \frac{1}{4} \). So, you'd have a quarter of the pizza left.

Common Denominator

Finding a common denominator is a fundamental step when working with fractions, particularly when adding or subtracting them. Think of a common denominator as a shared language between two fractions. Without a shared language, communication is difficult; the same applies to fractions. Without a common denominator, they can't be combined.

To find a common denominator, look for the smallest number that both denominators can divide into evenly. It's like looking for the smallest ground where all parties can meet. For example, to combine \( \frac{3}{8} \) and \( \frac{5}{12} \), the smallest common ground is 24 because it's the least number that both 8 and 12 can divide into. Convert both fractions to have 24 as the denominator, resulting in \( \frac{9}{24} \) and \( \frac{10}{24} \). Now, with a shared language, you can add or subtract them directly.

Algebraic Expressions

Algebraic expressions are like puzzles; they contain numbers, variables (like x or y), and operations (such as addition or subtraction), all of which work together to represent a value. These expressions are like recipes that tell you exactly what ingredients and steps you need to reach the final dish, or in this case, a solution.

An expression such as \( 12 - \frac{4}{x+2} \) is a combination of whole numbers and a fraction, where the variable x is part of the denominator in the fractional part. To simplify, treat the whole number as a fraction with a denominator of 1, just as you would measure all ingredients in the same units before combining them in a recipe.

When dealing with algebraic expressions, always perform operations in the correct order: parentheses, exponents, multiplication and division, addition and subtraction (PEMDAS/BODMAS). In our case, since subtraction is the target, we make the denominators match before combining the terms. The aim is to simplify the expression to its lowest form, just as you simplify a recipe by combining similar ingredients.