Problem 33 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The correct symbol is \( \neq \).

Step 1: Simplify the Expression on the Left

We start with the expression \( \frac{ab + ac}{a} \). To simplify, we divide each term in the numerator by \(a\). Thus, it becomes \( \frac{ab}{a} + \frac{ac}{a} = b + c \).

Step 2: Rewrite the Entire Equation

Replace the simplified left-hand side from Step 1 into the entire expression. The expression becomes \( b + c \square b + ac \).

Step 3: Analyze Possible Solutions

Consider when the expression \( b + c \) is equal to or not equal to \( b + ac \). Clearly, \( b + c eq b + ac \) for all real numbers \(a\), \(b\), and \(c\) because \(c\) is not necessarily equal to \(ac\).

Step 4: Choose the Correct Symbol

Since \(b + c \) is not the same as \(b + ac\) in general, the correct symbol to make the statement true for all real numbers is \( eq \). Thus, we write \( \frac{ab + ac}{a} eq b + ac \).

Key Concepts

Algebraic Simplification

Algebraic simplification is an essential skill in precalculus equations. It involves reducing expressions to their simplest form by performing valid operations. In this exercise, we began with the expression \( \frac{ab + ac}{a} \). The goal was to make this as simple as possible.To simplify, we focused on each part of the expression. The numerator \( ab + ac \) needed to be divided by \( a \). This allowed us to break it into two separate fractions:

• \( \frac{ab}{a} \)
• \( \frac{ac}{a} \)

Performing the division, \( \frac{ab}{a} \) simplifies to \( b \), and \( \frac{ac}{a} \) simplifies to \( c \). Adding the two results gives us the simplified form \( b + c \). This step is crucial because it sets the stage for analyzing the whole statement by making the expression easier to work with.Breaking down complex expressions into simpler parts not only helps in understanding but also makes the subsequent steps more manageable.

Equation Solving

Solving equations involves finding the correct symbol or value to make an equation true. In our exercise, the goal was to identify whether \( b+c \) is equal to or not equal to \( b+ac \).Once we simplified the initial expression to \( b+c \), the next task was to compare it with the expression \( b+ac \). This is where you determine the relationship between both sides.

• If both sides are exactly the same for all possible values, you would use the equality symbol (=).
• If they are not the same, you use the inequality symbol (\( eq \)).

In this case, \( b+c \) and \( b+ac \) are not equal in all circumstances since \( c \) is not equal to \( ac \) for every value of \( a \), \( b \), and \( c \). Therefore, it was necessary to choose the inequality symbol (\( eq \)) to ensure the equation statement holds true universally.

Real Numbers

Real numbers play a vital role in mathematics by providing a broad spectrum of values through which calculations and expressions operate. In the context of this exercise, real numbers include all values for \( a \), \( b \), and \( c \) that make our equation valid.Understanding real numbers involves grasping that they include all rational and irrational numbers.

• Rational numbers are those that can be expressed as fractions.
• Irrational numbers cannot be written as simple fractions.

Given the exercise's requirement for the equation to be true universally, utilizing real numbers ensures that the simplification and any relationships observed apply broadly across all potential values that \( a \), \( b \), or \( c \) might take. This universality is a key aspect of working with equations in precalculus, as it highlights the necessity of considering all possible scenarios and not just specific instances.

Mathematical Symbols

Mathematical symbols are crucial in conveying precise relationships between different parts of an equation. In this exercise, the responder was challenged to choose between the symbols \( = \) and \( eq \) to make a statement true.Each symbol carries distinct meaning:

• \( = \) indicates equivalence, signifying that the expressions on either side are identical in value or form.
• \( eq \) denotes inequality, showing that the two expressions are different.

The correct usage of mathematical symbols is central to effectively communicating mathematical ideas. Here, \( eq \) was chosen because the simplified expression, \( b+c \), could not equal \( b+ac \) for all real numbers. This choice directly impacts the truth of the statement across all conditions and is a testament to their foundational role in mathematical logic and reasoning. Such understanding helps one to interpret and construct equations and expressions accurately in the future.