Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation.
[tex](x-7)(x+2)\ \textgreater \ 0[/tex]
Answer (1)
Find the critical points by setting each factor to zero: x = 7 and x = − 2 .
Test the intervals ( − ∞ , − 2 ) , ( − 2 , 7 ) , and ( 7 , ∞ ) to determine where the inequality holds.
The inequality holds for ( − ∞ , − 2 ) and ( 7 , ∞ ) .
Express the solution in interval notation: ( − ∞ , − 2 ) ∪ ( 7 , ∞ ) .
Explanation
Understanding the Problem We are given the polynomial inequality 0"> ( x − 7 ) ( x + 2 ) > 0 . Our goal is to solve this inequality, represent the solution set on a number line, and express it in interval notation.
Finding Critical Points To solve the inequality, we first find the critical points by setting each factor to zero: x − 7 = 0 and x + 2 = 0 .
Determining Intervals Solving for x , we get the critical points x = 7 and x = − 2 . These points divide the number line into three intervals: ( − ∞ , − 2 ) , ( − 2 , 7 ) , and ( 7 , ∞ ) .
Testing Intervals Now, we test each interval to see where the inequality 0"> ( x − 7 ) ( x + 2 ) > 0 holds.
For the interval ( − ∞ , − 2 ) , let's choose a test point x = − 3 . Then, we have: 0"> ( − 3 − 7 ) ( − 3 + 2 ) = ( − 10 ) ( − 1 ) = 10 > 0 So, the inequality holds in this interval.
For the interval ( − 2 , 7 ) , let's choose a test point x = 0 . Then, we have: ( 0 − 7 ) ( 0 + 2 ) = ( − 7 ) ( 2 ) = − 14 < 0 So, the inequality does not hold in this interval.
For the interval ( 7 , ∞ ) , let's choose a test point x = 8 . Then, we have: 0"> ( 8 − 7 ) ( 8 + 2 ) = ( 1 ) ( 10 ) = 10 > 0 So, the inequality holds in this interval.
Expressing the Solution in Interval Notation The solution set consists of the intervals where the inequality holds. Therefore, the solution set in interval notation is ( − ∞ , − 2 ) ∪ ( 7 , ∞ ) .
Final Answer The solution to the inequality 0"> ( x − 7 ) ( x + 2 ) > 0 is the set of all x such that x < − 2 or 7"> x > 7 . In interval notation, this is written as ( − ∞ , − 2 ) ∪ ( 7 , ∞ ) .
Examples
Understanding polynomial inequalities helps in various real-world scenarios. For instance, if a company's profit is modeled by the expression ( x − 7 ) ( x + 2 ) , where x represents the number of units sold, solving the inequality 0"> ( x − 7 ) ( x + 2 ) > 0 tells us the sales levels at which the company makes a profit. This helps in making informed business decisions to ensure profitability.
