Problem 7 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression for \(x=4\). $$\frac{28}{x}$$

Step-by-Step Solution

Answer

The evaluated expression is 7.

Step 1: Understanding the expression

First, understand the given expression. It is \( \frac{28}{x} \), in this expression 28 is divided by x.

Step 2: Substituting the value of x

Plug in x = 4 into the expression. It becomes \( \frac{28}{4} \)

Step 3: Simplifying the expression

Perform the division to simplify the expression. The result is 7.

Key Concepts

Substitution in Algebra

Substitution is a fundamental concept in algebra where you replace a variable with its value. It's like being a detective, where variables are the clues, and the numbers they represent are the secrets you're trying to uncover. For instance, when given an algebraic expression like \( \frac{28}{x} \), and told that \( x=4 \), your job is to substitute the '4' wherever you see an 'x'. This is exactly the same as replacing a person's nickname with their real name so everyone knows who you're talking about.

Imagine you have five apples \( (x=5) \) and the expression \( 2x \), then substituting 'x' with '5' gives you \( 2 \times 5 = 10 \) apples. The process is simple yet powerful, enabling you to clear up the expression and prepare it for the next step, which is often simplification.

Expressions and Equations

Expressions and equations are the bread and butter of algebra. They're like sentences in the language of mathematics, conveying relationships between numbers and variables. An expression is a combination of numbers, variables, and operation symbols (like +, -, *, /) that stands for a single number value. Think of expressions as mathematical phrases that don't assert something completely, like the term 'sunny skies.'

Equations vs. Expressions

On the other hand, an equation is like a full sentence. It has an equal sign, \( = \), which acts like the verb 'is.' It tells you that two expressions on either side of the equal sign have the same value. So, while \( \frac{28}{x} \) is an expression, if you see something like \( \frac{28}{x} = 7 \), now you're dealing with an equation claiming that \( \frac{28}{x} \) is the same as 7.

Simplifying Algebraic Expressions

Simplifying is the process of making an algebraic expression as easy to understand as possible. It's like taking a complex recipe and breaking it down into just a few simple steps that anyone can follow. Simplification might involve combining like terms, reducing fractions, or applying the distributive property. The goal is to strip down the expression to its bare essentials while keeping its value unchanged.

Total clarity is crucial in simplification. For the expression \( \frac{28}{x} \), when \( x = 4 \), you substitute first to get \( \frac{28}{4} \), and then simplify it by carrying out the division, which leaves you with 7 – a much neater and more understandable result that stands crystal clear on its own.