Problem 154 - College Algebra Essentials

Question

What's wrong with this argument? Suppose \(x\) and \(y\) represent two real numbers, where \(x>y\). \(2>1 \quad\) This is a true statement \(2(y-x)>1(y-x) \quad\) Multiply both sides by \(y-x\) \(2 y-2 x>y-x \quad\) Use the distributive property. \(y-2 x>-x \quad\) Subtract \(y\) from both sides. \(y>x \quad\) Add \(2 x\) to both sides. The final inequality, \(y>x,\) is impossible because we were initially given \(x>y\).

Step-by-Step Solution

Answer

The error arises because the inequality sign is not reversed when both sides of the inequality are multiplied by a negative number. The corrected step should be \(2(y - x) < 1(y - x)\).

Step 1: Identify the initial true statement

We're given the true statement \(2 > 1.\)

Step 2: Analyze the algebraic manipulations

The algebraic manipulation consists of multiplying each side of the equation by \(y - x\), using the distributive property and, subtracting and adding terms.

Step 3: Pinpoint the flaw

The error arises from the second step, where both sides of the inequality \(2 > 1\) are multiplied by the expression \(y - x\). Since \(x > y\), obviously \(y - x < 0\). When we multiply both sides of an inequality by a negative number, the direction of the inequality sign should be reversed. However, this principle is not applied here when multiplying both sides by \(y - x\).

Step 4: Correct flawed step

The correct step should be \(2(y - x) < 1(y - x)\). Then the subsequent steps will not lead to the false conclusion \(y > x\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying mathematical expressions to solve problems or to understand relationships between different variables. This process often includes adding, subtracting, multiplying, or dividing both sides of an equation or inequality. By carefully manipulating the expressions, we can transform them while maintaining their inherent truths.

In the context of the exercise, algebraic manipulation was used in multiple steps. First, the expression was transformed by multiplying both sides of the inequality by a common factor, and then adding or subtracting terms from both sides. Each of these steps aims to reach a simplification, but requires careful observation of rules to avoid errors.

Distributive Property

The distributive property is a fundamental rule in algebra that allows us to multiply a single term by terms within parentheses. Put simply, it lets us distribute a factor across a sum or difference. Mathematically, it's stated as: \[a(b + c) = ab + ac\]

In our exercise, the distributive property was applied to expand expressions like \(2(y-x)\) into \(2y-2x\). This is a crucial step that simplifies expressions and sets the stage for further manipulation. However, always remember to check subsequent operations after applying the distributive property, ensuring all terms remain correctly transformed. This understanding helps avoid potential mistakes in later calculations.

Inequality Signs

Understanding inequality signs is crucial when dealing with inequalities. There are several inequality signs: greater than \(>\), less than \(<\), greater than or equal to \(\geq\), and less than or equal to \(\leq\). These symbols are used to compare the sizes of two values or expressions. Unlike equations that show equality, inequalities indicate a range of possible solutions.

In our scenario, we start with \(x > y\). As the expression changes during the solution steps, it's critical to keep track of these inequalities. Any operations on inequalities, like adding or subtracting a number from both sides, typically preserve the inequality sign. However, failing to correctly apply changes—such as flipping the sign when multiplying by a negative—can lead to incorrect conclusions, which happened here.

Multiplying Both Sides of an Inequality

When multiplying both sides of an inequality by a number, special care is required compared to equations. The rule is that if you multiply or divide both sides of an inequality by a positive number, the inequality remains the same. Conversely, when a negative number is involved, the inequality sign must be reversed.

In the given exercise, the critical mistake was failing to reverse the inequality sign after multiplying by \(y-x\). Given \(x > y\), \(y-x\) is negative, necessitating a reversal of the inequality sign once multiplication happens. This oversight led to a false conclusion. Always verify the nature of factors being multiplied or divided in inequalities—never overlook the sign change if they are negative.