Problem 3 - Elementary Algebra

Question

For the following problems, find the domain of each rational expression. $$ \frac{x+1}{2 x-5} $$

Step-by-Step Solution

Answer

Answer: The domain of the rational expression is all real numbers except \(x = \frac{5}{2}\), which can be written in interval notation as \((-\infty, \frac{5}{2}) \cup (\frac{5}{2}, \infty)\).

Step 1: Identify the denominator

In this rational expression, we can see the denominator is \((2x - 5)\).

Step 2: Set the denominator equal to zero

To find when the expression is undefined, set the denominator equal to zero: $$ 2x - 5 = 0 $$

Step 3: Solve for x

Solve the equation to find the value of x that makes the denominator zero: $$ 2x = 5 \\ x = \frac{5}{2} $$

Step 4: Determine the domain

The expression is undefined when \(x = \frac{5}{2}\). Therefore, the domain of the rational expression will be all real numbers except this value. In interval notation, this can be written as: $$ \text{Domain} = (-\infty, \frac{5}{2}) \cup (\frac{5}{2}, \infty) $$

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Just like with regular fractions, the key factor that we must remember is that the denominator cannot be zero because division by zero is undefined. To simplify or work with rational expressions, we often factor polynomials, cancel common factors, and simplify the expression.

A rational expression can take on every value except for where the denominator equals zero, since that would make the expression undefined. It's essential for students to identify the denominator first and then determine which values, if any, would cause it to become zero. When we find these values, we can say these are the restrictions on the domain of the expression. Understanding the limitations on the domain is crucial when working with rational expressions.

Solving for x

Solving for x is a fundamental skill in algebra that allows us to find the value(s) of the variable that satisfy an equation. When we are given a rational expression, and we need to find its domain, we solve for x by setting the denominator equal to zero and finding the x-values that we need to exclude from the domain. This process typically involves:

• Rearranging the equation such that x is on one side and all other terms are on the opposite side.
• Performing algebraic operations such as adding, subtracting, multiplying, dividing, and factoring.
• Isolating x and determining its value or values.

These steps help us determine where the rational expression is valid or undefined, which directly relates to the domain of the expression.

Interval Notation

Interval notation is a way of writing subsets of the real number line. It's a concise method used to describe domains and ranges of functions, including rational expressions. Let's take a closer look:

• An open interval \(a, b\) includes all the numbers between a and b but not a and b themselves. It's represented by parentheses.
• A closed interval \[a, b\] includes all the numbers between a and b along with a and b themselves. It's represented by square brackets.
• To signify that an interval extends to infinity, we use \(\infty\) or \-\infty\), and because infinity is not a number we can reach, these intervals are always open.

In the case of the exercised rational expression, since x cannot be \(\frac{5}{2}\), we use parenthesis around this value to exclude it from the domain, resulting in \(-\infty, \frac{5}{2}\) \cup (\frac{5}{2}, \infty\). This shows all real numbers are included except for \(\frac{5}{2}\).