Problem 73 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. \(\frac{x}{2}-\frac{x}{4}+4=x+4\)

Step-by-Step Solution

Answer

The given equation is true for all real numbers.

Step 1: Simplify the Left-hand Side (LHS)

First, simplify the left-hand side of the equation. Here, \( \frac{x}{2} - \frac{x}{4} + 4 \) needs to be simplified. To add/subtract fractions, they must have a common denominator. In this case, they do - it's 4. We can then rewrite \( \frac{x}{2} \) as \( \frac{2x}{4} \) . Hence, \( \frac{2x}{4} - \frac{x}{4} + 4 = \frac{x}{4} + 4 \)

Step 2: Simplify the Right-hand Side (RHS)

The right-hand side of the equation \( x + 4 \) is already simplified as it can be.

Step 3: Compare the Simplified LHS and RHS

After simplifying both sides, we find that they are equal, i.e., \( \frac{x}{4} + 4 = x + 4 \). This suggests that the equation is true for all real numbers, as the terms are identical.

Key Concepts

Fractions in Algebra

When dealing with fractions in algebra, it's pivotal to remember that they behave similarly to whole numbers in operations, following the same rules for addition, subtraction, multiplication, and division. A stumbling block for many students comes in the form of adding and subtracting algebraic fractions.

To perform these operations, we must ensure the fractions have a common denominator, just as we find a common base when working with exponents. Take the equation \(\frac{x}{2}-\frac{x}{4}+4=x+4\). Here, the denominators are 2 and 4. Multiplying the numerator and denominator of the first fraction by 2 gives us a new equivalent fraction \(\frac{2x}{4}\), which aligns with the denominator of the second fraction. With like denominators, combining terms becomes straightforward, leading to the consolidation of the variable terms.

Equation Simplification

Simplification of an equation is essential to isolate the variable and solve for unknown values. The main goal is to combine like terms and reduce the equation to its simplest form, making it easier to solve. In our example, we simplified the left-hand side by combining fractions with a common denominator and then adding the whole number.

This leaves us with a much more manageable expression \(\frac{x}{4} + 4\). Simplification can also involve expanding expressions, factoring, and canceling out terms. While simplification on the right-hand side wasn't necessary in our exercise, in other cases, distributing terms or combining like terms could be crucial for obtaining a clear path to the solution.

Common Denominators

Identifying common denominators is crucial for managing fractions within equations. The easiest way to find a common denominator is to use the least common multiple (LCM) of the denominators. In the given example, we found that 4 is the least common denominator for 2 and 4, which allows us to rewrite the fractions so that each has the same denominator.

The process ensures we're comparing like terms, which is a necessity in algebra. Once fractions have a common base, we can combine the numerators with confidence that we're not altering the value of the expressions. Additionally, the use of common denominators simplifies the path toward identifying cases of no solution or identity equations.

No Solution and Identity Equations

During the process of solving equations, you might encounter situations where the equation simplifies to an untrue statement like \(5=3\), indicating no solution exists, or to a true statement like \(4=4\) for all real numbers, which signifies we have an identity equation.

In the exercise \(\frac{x}{2}-\frac{x}{4}+4=x+4\), after simplification, the two sides of the equation are identical, leading us to conclude that the equation is true no matter what real number \(x\) is. This is a classic identity equation because it holds true universally, unlike an equation with no solution, which represents a contradiction in terms.