What is the solution set to the inequality $5(x-2)(x+4)>0$?
A. {x | x<-4 or x>2}
B. {x | x>-4 and x<2}
C. {x | x<-2 or x>4}
D. {x | x>-2 or x<4}

Divide both sides of the inequality by 5: 0"> ( x − 2 ) ( x + 4 ) > 0 .
Find critical points by setting each factor to zero: x = 2 and x = − 4 .
Test intervals ( − ∞ , − 4 ) , ( − 4 , 2 ) , and ( 2 , ∞ ) to find where the inequality holds.
The solution set is 2\}}"> { x ∣ x < − 4 or x > 2 } ​ .

Explanation

Understanding the Problem We are given the inequality 0"> 5 ( x − 2 ) ( x + 4 ) > 0 and we need to find the solution set for x .

Simplifying the Inequality First, divide both sides of the inequality by 5. Since 5 is a positive number, the inequality sign remains the same: 0"> ( x − 2 ) ( x + 4 ) > 0

Finding Critical Points Next, we find the critical points by setting each factor to zero: x − 2 = 0 A rr x = 2 x + 4 = 0 A rr x = − 4

Creating Intervals Now, we create a number line and mark the critical points − 4 and 2 . These points divide the number line into three intervals: ( − ∞ , − 4 ) , ( − 4 , 2 ) , and ( 2 , ∞ ) . We will test a value from each interval to determine where the inequality 0"> ( x − 2 ) ( x + 4 ) > 0 holds.

Testing Interval ( − ∞ , − 4 )

For the interval ( − ∞ , − 4 ) , let's test x = − 5 :
0"> ( − 5 − 2 ) ( − 5 + 4 ) = ( − 7 ) ( − 1 ) = 7 > 0 Since the result is greater than 0, the inequality holds for x < − 4 .

Testing Interval ( − 4 , 2 )

For the interval ( − 4 , 2 ) , let's test x = 0 :
( 0 − 2 ) ( 0 + 4 ) = ( − 2 ) ( 4 ) = − 8 < 0 Since the result is less than 0, the inequality does not hold for − 4 < x < 2 .

Testing Interval ( 2 , ∞ )

For the interval ( 2 , ∞ ) , let's test x = 3 :
0"> ( 3 − 2 ) ( 3 + 4 ) = ( 1 ) ( 7 ) = 7 > 0 Since the result is greater than 0, the inequality holds for 2"> x > 2 .

Final Solution The solution set is the union of the intervals where the inequality holds, which are x < − 4 and 2"> x > 2 . Therefore, the solution set is: 2\}"> { x ∣ x < − 4 or x > 2 }


Examples
Understanding inequalities is crucial in various real-world scenarios. For instance, consider a manufacturing company that needs to maintain a certain profit margin. If the cost of production is represented by C ( x ) and the revenue by R ( x ) , the company needs to ensure that 0"> R ( x ) − C ( x ) > 0 to make a profit. Solving such inequalities helps the company determine the optimal production levels to stay profitable. Similarly, in finance, inequalities are used to model investment returns and risk assessment, ensuring that potential gains outweigh potential losses.

Answered by GinnyAnswer | 2025-07-08