Question

Evaluate the expression for each value of $x$. (If not possible, state the reason.)$\frac{x+3}{x-3}$ (a) $x=3$ (b) $x=-3$

Short answer

The expression $\frac{x+3}{x-3}$ can't be evaluated when $x=3$ because it would involve division by zero which is undefined. The value of the expression when $x=-3$ is 0.

Step-by-step solution

Evaluate when $x=3$

Substitute $x=3$ into the expression $\frac{x+3}{x-3}$. This gives us $\frac{3+3}{3-3}$, simplifying to $\frac{6}{0}$. Since division by zero is undefined in mathematics, we say it's not possible to evaluate this expression when $x=3$.

Evaluate when $x=-3$

Substitute $x=-3$ into the expression $\frac{x+3}{x-3}$. This gives us $\frac{-3+3}{-3-3}$, which simplifies to $\frac{0}{-6}$. The value of any number divided by zero is 0. So the value of the expression when $x=-3$ is 0.

Key concepts

Division by Zero

When working with algebraic expressions, students often encounter the rule that division by zero is not allowed. This isn't just a random prohibition; it stems from the fundamental properties of numbers and division. Division is essentially the inverse of multiplication. For any number divided by another, there is an implied question: 'How many times does the divisor fit into the dividend?' For example, in the equation $\frac{6}{3}=2$, the number 3 fits into 6 exactly two times.

However, when you attempt to divide by zero, you run into a logical inconsistency. Let's imagine you have $\frac{6}{0}$. The question being asked is 'How many times does 0 fit into 6?' The answer is indeterminate because 0 cannot 'fit' into 6—or any number—any number of times; 0 multiplied by any number is always 0. Thus, $\frac{a}{0}$, where $a$ can be any number, doesn't have an answer that makes sense within our number system. It's not that the answer is zero; it's that the question itself doesn't have an answer that fits our understanding of division. This is why in mathematics, any division by zero is considered undefined.

Undefined Mathematical Expressions

Undefined mathematical expressions go beyond just division by zero. They encompass any situation where an operation cannot be completed according to the commonly accepted rules of arithmetic or algebra. Division by zero is a prime example because, as we just discussed, it does not result in a meaningful outcome.

There are also other types of undefined expressions. For instance, taking the square root of a negative number in the realm of real numbers is undefined because no real number squared will give a negative result. In algebra, sometimes expressions will become undefined due to the specific values that variables take on. An example can be seen in rational expressions—fractions that contain a variable in the denominator—where assigning a value to the variable that results in a zero denominator makes the expression undefined.

It is crucial for students to be aware of these undefined conditions to avoid errors in their calculations and to understand the limitations of certain operations. Recognizing when an expression is undefined is an essential skill in ensuring mathematical reasoning is accurate and meaningful.

Substituting Values in Algebraic Expressions

Substituting values into algebraic expressions is a fundamental skill in algebra. This process involves replacing the variables with specific numbers and then performing the standard operations of arithmetic: addition, subtraction, multiplication, and division. Substitution allows students to evaluate expressions at particular values to find a numerical result.

To correctly substitute values, it's important to follow these steps:

• Identify the variables in the expression.
• Determine the values that are to be substituted in place of the variables.
• Replace each occurrence of the variable with the chosen value.
• Carefully perform the arithmetic operations, paying particular attention to the order of operations.

Occasionally, as seen in the exercise where substituting $x=3$ into the expression $\frac{x+3}{x-3}$ results in $\frac{3+3}{3-3}$ or $\frac{6}{0}$, substitution can lead to an undefined expression. When this happens, the correct step is to recognize the limitation and simply state that the expression cannot be evaluated for that particular value of the variable. Properly understanding substitution and the potential for undefined results is essential for solving algebraic expressions confidently and accurately.