Problem 39 - 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. $$ \frac{a-b}{b-a} \square-1 $$

Step-by-Step Solution

Answer

Replace \( \square \) with \( = \).

Step 1: Simplify the Expression

We start by simplifying the expression \( \frac{a-b}{b-a} \). Notice that the denominator \( b-a \) is just the negative of the numerator \( a-b \). Therefore, we can rewrite the expression as:\[ \frac{a-b}{b-a} = \frac{a-b}{-(a-b)} = -1 \]

Step 2: Compare the Simplified Expression to -1

Now we need to compare the simplified expression \(-1\) to the number \(-1\). Since both are numerically equal, we can replace \( \square \) with the symbol \( = \).

Step 3: Conclusion

In this particular exercise, for all real numbers \( a, b \), the statement becomes true when we use \( = \), since the two expressions are equal for any values of \( a \) and \( b \) where the denominator \( b-a \) is not zero.

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves making expressions easier to understand or work with. When we simplify an expression, our goal is to make it as straightforward and concise as possible by eliminating unnecessary components.

In the case of the expression \( \frac{a-b}{b-a} \), simplification involves recognizing that both the numerator \( a-b \) and denominator \( b-a \) are equivalent but opposite in sign. Thus, \( b-a \) can be rewritten as \( -(a-b) \). This allows us to express the entire fraction as \( \frac{a-b}{-(a-b)} = -1 \).

Here are a few tips when simplifying expressions:

• Look for common factors in numerators and denominators.
• Be attentive to the signs—negatives can often be key in simplification.
• Rewriting expressions in equivalent forms can reveal simplification opportunities.

Rational Expressions

Rational expressions are essentially fractions in which both the numerator and the denominator are polynomials. Much like numbers, these expressions can often be simplified or manipulated using arithmetic operations.

In a rational expression like \( \frac{a-b}{b-a} \), it is crucial to explore the relationships between the terms. Recognizing equivalent terms (such as \( a-b \) and \( -b+a \)) plays a pivotal role in simplifying rational expressions.

Always ensure:

• The denominator never equals zero, as division by zero is undefined in mathematics.
• Use rules of factoring and division to reduce expressions to their simplest form.

By practicing and understanding these principles, you become adept at handling more complex algebraic problems.

Real Numbers

Real numbers encompass a wide range of numbers that include integers, fractions, and irrational numbers. They form the backbone of most algebraic work, providing the values we plug into equations and expressions.

For any mathematical statement or equation involving real numbers, it’s crucial to consider all possible values within this set, while acknowledging constraints such as the non-zero requirement in denominators.

When working with real numbers:

• Remember that real numbers can be both positive and negative, including zero.
• Equations involving expressions, as in the provided exercise, should consider properties of real numbers like closure, associativity, and distribution.
• Be wary of cases where operations are not defined, such as division by zero.

Real numbers play an essential role in ensuring that algebraic expressions can be evaluated across a wide spectrum, helping validate their truthfulness in exercises like replacing symbols in expressions.