Problem 4 - Elementary Algebra

Question

For the following problems, find the domain of each rational expression. $$ \frac{2 a+3}{7 a+5} $$

Step-by-Step Solution

Answer

Answer: The domain of the given rational expression is $a \in \mathbb{R} - \left\{-\frac{5}{7}\right\}$, which includes all real numbers except for $-5/7$.

Step 1: Identify the Denominator

In the given rational expression, identify the denominator which is: $$ 7a + 5 $$

Step 2: Set the Denominator Not Equal to Zero

In order for the expression to be defined, the denominator cannot be equal to zero. Set the denominator not equal to zero and solve for 'a': $$ 7a + 5 \neq 0 $$

Step 3: Solve for the Variable 'a'

Subtract 5 from both sides of the inequality and then divide by 7: $$ 7a \neq -5 \\ a \neq -\frac{5}{7} $$

Step 4: State the Domain

Since the denominator is defined for all values of 'a' except for -5/7, the domain of the rational expression is: $$ a \in \mathbb{R} - \left\{-\frac{5}{7}\right\} $$ This means that the domain includes all real numbers except for -5/7.

Key Concepts

Understanding Rational Expressions

A rational expression is much like a fraction, but instead of being made up of just numbers, it can include variables and algebraic expressions. Essentially, it’s an expression of the form:

• \( \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials.

A crucial aspect of rational expressions is that they are undefined when their denominators are zero. This is because division by zero is impossible in mathematics.
To work with rational expressions effectively, it's essential to identify when they are undefined, which involves understanding the role of the denominator. By inspecting the denominator, you can determine the values that make the expression undefined and thus influence its domain.
Solving problems involving rational expressions requires attention to detail, especially when simplifying or performing operations like addition, subtraction, multiplication, or division.
Understanding and identifying valid operations within the domain play a significant role in managing these expressions effectively.

The Role of the Denominator in Algebra

The denominator in any expression plays a vital role, mainly as it determines the values for which the expression is undefined. In algebra, and specifically in the study of rational expressions, the denominator is key to understanding where the expression is valid.
For example, in the expression \( \frac{2a+3}{7a+5} \), the denominator is \( 7a+5 \).

• To ensure the expression is valid, set \( 7a+5 eq 0 \).
• This prevents division by zero, a fundamental rule in mathematics.

By solving \( 7a+5 eq 0 \), you find the specific values of the variable that must be excluded from the solution set. This process not only ensures expressions remain defined but also helps in defining the domain of these expressions.

Determining the Domain in Mathematics

The domain of a mathematical expression is the set of all possible input values (often \( x \) or another variable) that make the expression work. When dealing with rational expressions, finding the domain is a critical step because it tells you for which values the expression is true and can be safely evaluated.
To determine the domain:

• Identify the denominator.
• Set it not equal to zero to avoid undefined expressions.
• Solve this inequality to find any restricted values.
• The domain includes all real numbers except these restricted values.

For the expression \( \frac{2a+3}{7a+5} \), the restricted value is \( a eq -\frac{5}{7} \).
Therefore, the domain can be expressed as all real numbers except \(-\frac{5}{7}\), or in notation: \( a \in \mathbb{R} - \{-\frac{5}{7}\} \).
Recognizing these restrictions is vital for solving and simplifying rational expressions accurately in mathematics.