Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation.
[tex]\frac{x+1}{x+4}\ \textgreater \ 0[/tex]
Answer (1)
Find critical values: x = − 1 and x = − 4 .
Test intervals ( − ∞ , − 4 ) , ( − 4 , − 1 ) , and ( − 1 , ∞ ) .
Determine intervals where the inequality 0"> x + 4 x + 1 > 0 holds true.
Express the solution in interval notation: ( − ∞ , − 4 ) ∪ ( − 1 , ∞ ) .
Explanation
Finding Critical Values We are given the rational inequality 0"> x + 4 x + 1 > 0 . To solve this inequality, we first find the critical values by setting the numerator and denominator equal to zero.
Determining Intervals Setting the numerator equal to zero, we have x + 1 = 0 , which gives x = − 1 . Setting the denominator equal to zero, we have x + 4 = 0 , which gives x = − 4 . These critical values divide the number line into three intervals: ( − ∞ , − 4 ) , ( − 4 , − 1 ) , and ( − 1 , ∞ ) .
Testing Intervals Now, we choose a test value from each interval and plug it into the inequality 0"> x + 4 x + 1 > 0 to determine if the inequality is true or false in that interval.
Interval ( − ∞ , − 4 ) : Let's choose x = − 5 . Then 0"> − 5 + 4 − 5 + 1 = − 1 − 4 = 4 > 0 . The inequality is true in this interval.
Interval ( − 4 , − 1 ) : Let's choose x = − 2 . Then − 2 + 4 − 2 + 1 = 2 − 1 < 0 . The inequality is false in this interval.
Interval ( − 1 , ∞ ) : Let's choose x = 0 . Then 0"> 0 + 4 0 + 1 = 4 1 > 0 . The inequality is true in this interval.
Writing the Solution Set Since the inequality is strict ( "> > ), the critical values are not included in the solution set. Therefore, the solution set consists of the intervals ( − ∞ , − 4 ) and ( − 1 , ∞ ) .
Final Solution The solution set in interval notation is ( − ∞ , − 4 ) ∪ ( − 1 , ∞ ) .
Examples
Rational inequalities are used in various fields, such as economics and physics, to model constraints and optimization problems. For example, in economics, a company might use a rational inequality to determine the price range for a product that ensures a certain profit margin. Suppose the profit margin is given by Q ( x ) P ( x ) , where P ( x ) represents the profit and Q ( x ) represents the quantity sold. The company wants to ensure that k"> Q ( x ) P ( x ) > k , where k is a desired profit level. Solving this inequality helps the company find the range of quantities x that meet the profit goal.
