Problem 72 - 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. \(\frac{x}{4}+3=\frac{x}{4}\)

Step-by-Step Solution

Answer

The given equation has no solution.

Step 1: Simplify both sides

On the left side of the equation there is the expression \(\frac{x}{4} + 3\), and on the right side there is just \(\frac{x}{4}\).

Step 2: Check if identical

After simplifying, we notice that the expressions on either side of the equality are not equal to each other, because the left side contains an addition of 3, while the right side does not.

Step 3: Transition and simplification

Subtract \(\frac{x}{4}\) from both sides to get the equation \(3 = 0\).

Step 4: Interpret the result

The equation \(3 = 0\) is clearly not possible. This implies that the original equation does not have any solutions, because it suggests a false statement.

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is fundamental when solving equations. An algebraic expression is a mathematical phrase that can include numbers, variables (like x), and operation signs (such as plus, minus, multiply, and divide). For example, \( \frac{x}{4} + 3 \) is an algebraic expression where \( x \) is a variable, \( 4 \) and \( 3 \) are constants, and division and addition are the operations connecting them.

When solving an equation, we aim to isolate the variable and find its value. However, in some cases, the process of simplification reveals the nature of the expression—whether it can be solved for some value of the variable or if it's an equation with no solution. In our example, simplification leads us to a statement that is not possible, showing the primary importance of understanding and manipulating algebraic expressions properly.

No Solution Equations

Equations that have no solution can be puzzling at first, but recognizing them is a key skill in algebra. These are also known as inconsistent equations. A no solution equation means that no real number exists that can satisfy the equation— in other words, the equation's statements are false for all real numbers. For example, after simplifying the given expression \( \frac{x}{4}+3=\frac{x}{4} \) and subtracting \( \frac{x}{4} \) from both sides, we get the equation \(3 = 0\), which is impossible as no value of x can make this statement true.

To identify such equations, simplify and attempt to isolate the variable. If you're left with a statement that does not hold true, like a constant not equal to itself, that's when you've discovered an equation with no solution.

Real Numbers

The concept of real numbers is a foundation in understanding various mathematical problems, including algebraic equations. The set of real numbers includes all the numbers we commonly use, such as whole numbers, fractions, irrational numbers (like \( \pi \) or \( \sqrt{2} \)), and negatives of these. One of the properties of real numbers is that they can be ordered on a number line.

When analyzing equations, we typically look for solutions within the real numbers. However, not all equations, as discussed in 'No Solution Equations', will have a solution in this set. It’s crucial to remember that even though an equation may not have a solution within the set of real numbers, it doesn't mean there’s an error in the process. The equation itself is inherently unsolvable, such as in our original problem.