Problem 6 - Precalculus: Mathematics for Calculus

Question

Yes or No? If No , give a reason. (Disregard any value that makes a denominator zero.) (a) Is the expression \(\frac{3+a}{3}\) equal to \(1+\frac{a}{3} ?\) (b) Is the expression \(\frac{2}{4+x}\) equal to \(\frac{1}{2}+\frac{2}{x} ?\)

Step-by-Step Solution

Answer

(a) Yes; (b) No, different forms cannot align.

Step 1: Analyze Expression (a)

Consider the expression \( \frac{3+a}{3} \). We can rewrite this as \( \frac{3}{3} + \frac{a}{3} \). Since \( \frac{3}{3} = 1 \), the expression simplifies to \( 1 + \frac{a}{3} \).

Step 2: Compare Expression (a)

The expression \( \frac{3+a}{3} \) simplifies to \( 1 + \frac{a}{3} \) directly, which matches the expression provided as \( 1+\frac{a}{3} \). Therefore, the expressions are equal.

Step 3: Analyze Expression (b)

Consider the expression \( \frac{2}{4+x} \). There is no direct way to decompose or simplify this to form \( \frac{1}{2} + \frac{2}{x} \).

Step 4: Compare Expression (b)

Compare \( \frac{2}{4+x} \) and \( \frac{1}{2} + \frac{2}{x} \). By finding a common denominator, say \( (4+x)x \), and rewriting both, there's no simplification path that makes them equal, given the constraint \( x eq 0 \). Additionally, substitute values for \( x \) (avoid where \( 4+x = 0 \) or \( x = 0 \)), and the values don't lead to equivalence, proving they are unequal.

Key Concepts

Expression Simplification

Expression simplification involves the process of rewriting math expressions in a more compact or recognizable form while maintaining the expressions' original value. It aims to reduce complexity by breaking down expressions into more manageable parts.

For instance, take the expression \( \frac{3+a}{3} \). To simplify this, think of it as two separate fractions added together: \( \frac{3}{3} + \frac{a}{3} \).

• Begin by considering each part. \( \frac{3}{3} \) simplifies to 1, because any number divided by itself equals 1.
• The remaining part, \( \frac{a}{3} \), doesn't simplify further but is kept in fraction form.

When combined, these give \( 1 + \frac{a}{3} \), which yields a cleaner expression that's mathematically equivalent to the original but easier to work with.

Algebraic Equivalence

Algebraic equivalence refers to the concept where two expressions interchangeably share the same value, irrespective of how they might appear on the surface.

In our exercise, we analyzed if the given expressions were equivalent:
1. For expression \( \frac{3+a}{3} \), we found it to be equivalent to \( 1 + \frac{a}{3} \) upon simplification. It shows that the ordered manipulation of terms resulted in identical expressions, confirming equivalence.
2. On the other hand, \( \frac{2}{4+x} \) vs. \( \frac{1}{2} + \frac{2}{x} \) posed a different scenario. Attempting common denominators doesn’t result in equality, illustrating that even well-intended operations might not guarantee equivalence.

• To establish equivalence, simplified forms of each expression need identical outcomes.
• Always verify by substituting values to check practical correctness.

Denominator Constraints

A key aspect to keep in mind when working with rational expressions is handling denominator constraints. These determine the set of permissible values that a variable in the denominator can take.

For instance, expression \( \frac{2}{4+x} \) has a constraint: \( 4+x eq 0 \). Similarly, in any fraction, the denominator must not be zero, because division by zero is undefined.

• Always first identify and address potential zeros in the denominator when simplifying expressions.
• Analyze all scenarios where the denominator equals zero and restrict those from the set of possible solutions.

This practice ensures that your mathematical operations remain valid across all expected cases, reaffirming the integrity of rational expressions.