Welcome to our lesson on rational inequalities. A rational inequality is an inequality that contains one or more rational expressions. A rational expression is simply a fraction where both the numerator and denominator are polynomials. When we combine this with inequality symbols like less than, greater than, less than or equal to, or greater than or equal to, we get a rational inequality. Let's look at some examples to make this concept clear.
Now let's break down the key components of rational inequalities. Every rational inequality has three essential parts. First, we have the rational expression, which is a fraction containing polynomials in both the numerator and denominator. Second, we have the inequality symbol, which can be less than, greater than, less than or equal to, or greater than or equal to. Third, we have a comparison value, which could be zero, a number, or even another expression. It's crucial to remember that domain restrictions apply wherever the denominator equals zero, as division by zero is undefined.
Let's learn how to solve rational inequalities step by step. The general method involves four key steps. First, move all terms to one side of the inequality. Second, find the critical points by setting the numerator and denominator equal to zero separately. For our example, x plus one equals zero gives us x equals negative one, and x minus two equals zero gives us x equals two. Third, test the sign of the expression in each interval created by these critical points. Finally, write the solution by including all intervals where the inequality is satisfied. The number line shows us that the expression is positive when x is less than negative one or x is greater than two.
Rational inequalities come in various types and have many real-world applications. Simple rational inequalities involve basic fractions with polynomial expressions, while complex ones may have multiple rational expressions or higher-degree polynomials. These inequalities are widely used in economics for cost-benefit analysis, in physics for solving rate problems, in engineering for efficiency calculations, and in medicine for dosage optimization. The graph shows a typical rational function with vertical and horizontal asymptotes. Remember these key tips: always check domain restrictions where denominators equal zero, test points in each interval to determine the sign, and be careful about inequality direction when multiplying or dividing by negative values.
To summarize what we have learned about rational inequalities: They are mathematical expressions containing fractions with polynomials that use inequality symbols. The solution process involves finding critical points where numerators and denominators equal zero, then testing intervals to determine where the inequality holds true. Remember that domain restrictions apply wherever denominators equal zero. These inequalities have practical applications in economics, physics, and engineering. Always verify your solutions by substituting back into the original inequality.