Problem 34 - 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. $$ \frac{a b+a c}{a} \square b+c $$

Step-by-Step Solution

Answer

The correct symbol is \( = \).

Step 1: Analyze the given expression

We are given the expression \( \frac{ab + ac}{a} \square b + c \). The task is to replace \( \square \) with either \( = \) or \( eq \) to make the statement true for all real numbers \( a, b, c \), where the expression is defined (meaning \( a eq 0 \)).

Step 2: Simplify the left side of the equation

Simplify the left side of the expression \( \frac{ab + ac}{a} \). This can be broken into two separate fractions: \( \frac{ab}{a} + \frac{ac}{a} \). This simplifies to \( b + c \), since the \( a \) in the numerator and denominator cancel each other out.

Step 3: Compare both sides

Now, the expression becomes \( b + c \square b + c \). Both sides of the expression are exactly the same, meaning they will always be equal for any real numbers \( b \) and \( c \).

Step 4: Determine the correct symbol

Since \( b + c \) is always equal to \( b + c \), the correct symbol to use is \( = \). Thus, the expression \( \frac{ab + ac}{a} = b + c \) is true for all defined values of \( a \), \( b \), and \( c \).

Key Concepts

Simplification

Simplification involves reducing complex expressions into simpler forms, making them easier to understand and work with. Let's consider the expression:

• \( \frac{ab + ac}{a} \)

The simplification process helps us observe that we can separate this combined fraction into two distinct parts:

• \( \frac{ab}{a} + \frac{ac}{a} \)

Here, you can see that the variable \(a\) in both parts of the numerator and denominator cancels out when \( a eq 0 \):

• \( b + c \)

Breaking down the expression in such a manner not only makes it simpler but also shows the equality of both sides of the equation. Always look for opportunities to divide out common terms, as they often reveal the true and simpler form of your mathematical expressions.

Equality

Equality in mathematics signifies that two expressions are identical in value. In the exercise, we dealt with the expression:

• \( b + c \square b + c \)

We aimed to find whether a simple equality \( = \) or inequality \( eq \) sign could make this statement true.Upon simplifying the left side of the expression, we found:

• \( b + c = b + c \)

It is clear that both sides are exactly the same, so the sign that best fits is \( = \). This indicates that for all real numbers \(b\) and \(c\), the expression holds true. Recognizing equality helps us in understanding and affirming the relationships between mathematical expressions.

Real Numbers

Real numbers constitute a vast set of numbers including all rational and irrational numbers. They provide a comprehensive base for mathematical solutions.In this exercise, we focus on ensuring the expression \( \frac{ab + ac}{a} = b + c \) is valid for all real values of \( a, b, \) and \( c \), except when \( a = 0 \) (as this would make the expression undefined).

• **Rational numbers:** Numbers that can be expressed as fractions, including whole numbers and integers.
• **Irrational numbers:** Numbers that cannot be expressed as simple fractions, such as \( \pi \) or the square root of 2.

Working with real numbers ensures that our solutions are applicable in real-world scenarios. This broadness allows for flexibility and consistency in mathematical operations, providing us with a reliable foundation for solving any algebraic problem.