Problem 82 - College Algebra

Question

What is a rational inequality?

Step-by-Step Solution

Answer

A rational inequality is an inequality that involves a rational function, which is a function represented as the quotient of two polynomials. Its solution involves finding the range of possible values for the variable that satisfies the inequality, while ensuring the denominator is not zero.

Step 1: Definition

A rational inequality is an inequality that involves a rational function. A rational function is a function that can be represented as the quotient of two polynomials. In mathematical terms, if \(p(x)\) and \(q(x)\) are two polynomials, then \(r(x)=p(x)/q(x)\) is a rational function.

Step 2: Clarification on the inequality

So a rational inequality, is an inequality which involves a rational function. For example, let's take a rational function \(r(x) = p(x) / q(x)\). A rational inequality would then be an inequality that involves \(r(x)\), for instance, \(r(x) > 0\), \(r(x) < 0\), \(r(x) \geq 0\) or \(r(x) \leq 0\).

Step 3: Possible values of rational inequalities

Just like other types of inequalities, rational inequalities also indicate a range of possible values for x that satisfy the inequality. To find these possible values, we typically find the boundary points where the denominator \(q(x)\) is not equal to zero (since division by zero is undefined), and test the intervals between these boundary points to see if they satisfy the inequality.