Problem 66 - Introductory Algebra for College Students

Question

Why should restrictions on the variable in a rational equation be listed before you begin solving the equation?

Step-by-Step Solution

Answer

Upon solving a rational equation, restrictions for the variable must be listed beforehand to avoid coming up with solutions that would make the denominator equal to zero, which is undefined in mathematics.

Step 1: Understanding Rational Equations

Rational equations are equations including one or more rational expressions, which are algebraic expressions that include a fraction with a polynomial in the numerator and denominator. Since it's not allowed to divide by zero in math, there may be some values of the variable which render the rational equation undefined.

Step 2: Examining the Reason for Restrictions

Rational equations can become undefined if the value you substitute for the variable turns the denominator into zero, making division impossible. So, there may be some values that the variable can’t take on, and these are the 'restrictions' on the variable.

Step 3: Reason for Listing Restrictions First

At the start of solving a rational equation, it's beneficial to list the restrictions on the variable immediately. This is because a solution that makes the denominator equal to zero might still be mathematically valid in the general process of solving the equation. But in context of the starting equation, it's meaningless as it results in dividing by zero. By listing restrictions first, you ensure that you disregard any invalid solutions right away.