Solve the inequality and graph the solution set.
$\frac{x+9}{x-4}>0$
Answer (2)
Find the critical points by setting the numerator and denominator equal to zero: x + 9 = 0 and x − 4 = 0 , which gives x = − 9 and x = 4 .
Test the intervals ( − ∞ , − 9 ) , ( − 9 , 4 ) , and ( 4 , ∞ ) to determine where the inequality is satisfied.
The inequality is satisfied in the intervals ( − ∞ , − 9 ) and ( 4 , ∞ ) .
The solution set is ( − ∞ , − 9 ) ∪ ( 4 , ∞ ) . ( − ∞ , − 9 ) ∪ ( 4 , ∞ )
Explanation
Understanding the Inequality We are given the inequality 0"> x − 4 x + 9 > 0 . Our goal is to find all values of x that satisfy this inequality.
Finding Critical Points To solve this inequality, we first need to find the critical points. These are the values of x where the expression x − 4 x + 9 is either equal to zero or undefined. This occurs when the numerator or the denominator is equal to zero.
Calculating Critical Points Let's find the zeros of the numerator and the denominator:
Numerator: x + 9 = 0 ⇒ x = − 9
Denominator: x − 4 = 0 ⇒ x = 4
So, the critical points are x = − 9 and x = 4 .
Testing Intervals Now, we will test the intervals determined by these critical points on the number line: ( − ∞ , − 9 ) , ( − 9 , 4 ) , and ( 4 , ∞ ) .
Interval ( − ∞ , − 9 ) : Choose a test point, say x = − 10 . Then 0"> − 10 − 4 − 10 + 9 = − 14 − 1 = 14 1 > 0 So, the inequality is satisfied in this interval.
Interval ( − 9 , 4 ) : Choose a test point, say x = 0 . Then 0 − 4 0 + 9 = − 4 9 = − 4 9 < 0 So, the inequality is not satisfied in this interval.
Interval ( 4 , ∞ ) : Choose a test point, say x = 5 . Then 0"> 5 − 4 5 + 9 = 1 14 = 14 > 0 So, the inequality is satisfied in this interval.
Determining the Solution Set The solution set consists of the intervals where the inequality is satisfied. Therefore, the solution set is ( − ∞ , − 9 ) ∪ ( 4 , ∞ ) . Note that we use open intervals because the inequality is strict ( "> > ), so the critical points are not included in the solution.
Final Answer The solution to the inequality 0"> x − 4 x + 9 > 0 is x ∈ ( − ∞ , − 9 ) ∪ ( 4 , ∞ ) .
Examples
Understanding inequalities like this is useful in many real-world scenarios. For example, imagine you are managing a business and want to ensure your profit margin stays above a certain level. You can set up an inequality to model your revenue and costs, and then solve it to find the range of sales volumes that will keep your profit margin where you want it. This type of problem also appears in physics, engineering, and economics, where understanding the ranges of possible values is crucial for making informed decisions.
The inequality 0"> x − 4 x + 9 > 0 is solved by finding critical points and testing intervals. The solution set is ( − ∞ , − 9 ) ∪ ( 4 , ∞ ) , where the expression is positive. Therefore, values of x in these intervals will satisfy the inequality.
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