Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 33-40: Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true for all real numbers \(a, b\), \(c\), and \(d\), whenever the expressions are defined. $$ (a-b)-c \square a-(b-c) $$

Step-by-Step Solution

Answer

Replace \(\square\) with \(\neq\).

Step 1: Distribute Values

Expand the left side of the expression: \((a-b)-c = a-b-c\)

Step 2: Rearrange Terms on the Right

Expand the right side of the expression: \(a-(b-c) = a-b+c\)

Step 3: Compare Both Expressions

We compare the expressions from steps 1 and 2:- Left side: \(a-b-c\)- Right side: \(a-b+c\)These two expressions are not equal since the signs for the term \(c\) differ.

Step 4: Choose the Correct Symbol

Since \(a-b-c eq a-b+c\) for any real number \(c\), the correct symbol to use is \(eq\).

Key Concepts

Equations: Understanding Equality and Inequality

In mathematics, equations play a crucial role when it comes to expressing relationships between different quantities. An equation uses the symbol \( = \) to denote that two expressions are equal. Alternatively, the symbol \( eq \) is used when two expressions are not equal.

In the context of the given exercise, we deal with understanding whether two rearranged expressions are equal or not under any condition for real numbers. The challenge was to determine whether the two sides of the given algebraic statement can be equal for all possible values. This involves careful comparison and logical deduction to arrive at either \( = \) or \( eq \).

The step-by-step solution provided for the exercise helps break down this process by first fully expressing each side and then drawing a direct comparison. This logical reasoning helps students grasp the underlying principle of comparing algebraic expressions effectively to determine their equivalence or lack thereof.

Real Numbers: The Groundwork of Algebra

Real numbers are the foundation of many algebraic concepts. In algebra, real numbers include

• integers (like -3, 0, 2)
• rational numbers, which are numbers that can be expressed as a fraction, such as 1/2 or -5/7
• irrational numbers, which cannot be written as a simple fraction, such as \( \sqrt{2} \) or \( \pi \)

Real numbers are vital because they ensure that the operations performed in algebra are applicable to vast sets of numbers, covering all these types without exception.

In the given exercise, we are asked to consider the expressions with variables \(a, b, c,\) and \(d\) as placeholders for real numbers. This means any valid real number can be substituted into these expressions, making the task of ensuring equivalence or inequality applicable across an entire spectrum of numbers.

The universality of real numbers in algebra allows for comprehensive solutions and analyses because it guarantees that our findings are not limited by the type of numbers in use.

Expression Simplification: Making Problems Easier

Expression simplification is an essential skill in algebra that involves rewriting expressions in a more manageable form while retaining their original values.

The simplification process typically includes a few strategic steps:

• Distributing terms, including operations like addition and subtraction
• Combining like terms to reduce complexity
• Rearranging expressions to check for equality or solve problems

In our exercise, we saw how simplification allows us to unfold parentheses and logically structure the expressions. By distributing terms in both \((a-b)-c\) and \(a-(b-c)\), we could then directly compare the simplified forms \(a-b-c\) and \(a-b+c\).

Simplification not only makes expressions clearer but also exposes their true structure, helping determine relationships between different parts. This is particularly useful in identifying the correct symbol, \(eq\), to distinguish the inequality presented by the original expression in the exercise.