Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation. [tex]$\frac{x-9}{x+5}\ \textgreater \ 0$[/tex] Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is (9,-5). (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) B. The solution set is the empty set.
Answer (2)
The solution to the inequality x + 5 x − 9 > 0 is ( − ∞ , − 5 ) ∪ ( 9 , ∞ ) . This indicates that the expression is greater than zero in the intervals before -5 and after 9. The critical points -5 and 9 are not included in the solution set.
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Find the critical values by setting the numerator and denominator equal to zero: x = 9 and x = − 5 .
Test intervals ( − ∞ , − 5 ) , ( − 5 , 9 ) , and ( 9 , ∞ ) to determine where the inequality 0"> x + 5 x − 9 > 0 is true.
The inequality is true for ( − ∞ , − 5 ) and ( 9 , ∞ ) .
Express the solution set in interval notation: ( − ∞ , − 5 ) ∪ ( 9 , ∞ ) .
Explanation
Understanding the Problem We are given the rational inequality 0"> x + 5 x − 9 > 0 . Our goal is to solve this inequality and express the solution set in interval notation.
Finding Critical Values To solve the inequality, we first find the critical values by setting the numerator and denominator equal to zero:
Numerator: x − 9 = 0 ⇒ x = 9
Denominator: x + 5 = 0 ⇒ x = − 5
These critical values divide the number line into three intervals: ( − ∞ , − 5 ) , ( − 5 , 9 ) , and ( 9 , ∞ ) .
Testing Intervals Next, we choose a test value from each interval and plug it into the inequality 0"> x + 5 x − 9 > 0 to determine if the inequality is true or false in that interval.
Interval ( − ∞ , − 5 ) : Choose x = − 6 . Then 0"> − 6 + 5 − 6 − 9 = − 1 − 15 = 15 > 0 . The inequality is true.
Interval ( − 5 , 9 ) : Choose x = 0 . Then 0 + 5 0 − 9 = 5 − 9 < 0 . The inequality is false.
Interval ( 9 , ∞ ) : Choose x = 10 . Then 0"> 10 + 5 10 − 9 = 15 1 > 0 . The inequality is true.
Determining the Solution Set The solution set consists of the intervals where the inequality is true. Therefore, the solution set is ( − ∞ , − 5 ) ∪ ( 9 , ∞ ) . Note that we use parentheses because the inequality is strict ( 0"> > 0 ), so the critical values x = − 5 and x = 9 are not included in the solution set.
Final Answer The solution set is ( − ∞ , − 5 ) ∪ ( 9 , ∞ ) .
Examples
Rational inequalities are used in various real-world applications, such as determining the break-even point for a business. For example, a company's profit margin might be modeled by a rational function, and solving a rational inequality helps determine the sales volume needed to achieve a certain profit level. Understanding these inequalities is crucial for making informed business decisions and managing resources effectively.
