$(x-1)(x-4)(x-5) \leq 0
Answer (2)
Find the roots of the equation ( x − 1 ) ( x − 4 ) ( x − 5 ) = 0 , which are x = 1 , 4 , 5 .
Test the intervals ( − ∞ , 1 ) , ( 1 , 4 ) , ( 4 , 5 ) , and ( 5 , ∞ ) to determine where the expression ( x − 1 ) ( x − 4 ) ( x − 5 ) is negative or zero.
Include the roots 1 , 4 , 5 in the solution since the inequality is non-strict ( ≤ 0 ).
The solution set is ( − ∞ , 1 ] ∪ [ 4 , 5 ] .
Explanation
Understanding the Problem We are given the inequality ( x − 1 ) ( x − 4 ) ( x − 5 ) ≤ 0 . We need to find the values of x that satisfy this inequality.
Finding the Roots First, let's find the roots of the equation ( x − 1 ) ( x − 4 ) ( x − 5 ) = 0 . The roots are x = 1 , 4 , 5 . These are the points where the expression changes sign.
Creating a Number Line and Testing Intervals Now, let's create a number line and mark the roots 1 , 4 , 5 on it. We will test the intervals ( − ∞ , 1 ) , ( 1 , 4 ) , ( 4 , 5 ) , and ( 5 , ∞ ) to determine where the expression ( x − 1 ) ( x − 4 ) ( x − 5 ) is negative or zero.
Testing the Intervals
For x < 1 , let's pick x = 0 . Then ( 0 − 1 ) ( 0 − 4 ) ( 0 − 5 ) = ( − 1 ) ( − 4 ) ( − 5 ) = − 20 < 0 . Thus, the interval ( − ∞ , 1 ) is part of the solution.
For 1 < x < 4 , let's pick x = 2 . Then 0"> ( 2 − 1 ) ( 2 − 4 ) ( 2 − 5 ) = ( 1 ) ( − 2 ) ( − 3 ) = 6 > 0 . Thus, the interval ( 1 , 4 ) is not part of the solution.
For 4 < x < 5 , let's pick x = 4.5 . Then ( 4.5 − 1 ) ( 4.5 − 4 ) ( 4.5 − 5 ) = ( 3.5 ) ( 0.5 ) ( − 0.5 ) = − 0.875 < 0 . Thus, the interval ( 4 , 5 ) is part of the solution.
For 5"> x > 5 , let's pick x = 6 . Then 0"> ( 6 − 1 ) ( 6 − 4 ) ( 6 − 5 ) = ( 5 ) ( 2 ) ( 1 ) = 10 > 0 . Thus, the interval ( 5 , ∞ ) is not part of the solution.
Including the Roots Since the inequality is non-strict ( ≤ 0 ), we include the roots 1 , 4 , 5 in the solution. Therefore, the solution set is the union of the intervals where the inequality holds, including the roots.
Final Solution The solution set is ( − ∞ , 1 ] ∪ [ 4 , 5 ] .
Examples
Understanding inequalities like this is crucial in many real-world scenarios. For example, in engineering, you might need to determine the range of values for a parameter that keeps a system stable. Similarly, in economics, you might analyze price ranges that ensure profitability. This problem demonstrates how to find the intervals where a given condition is met, a fundamental skill in various fields.
To solve the inequality ( x − 1 ) ( x − 4 ) ( x − 5 ) ≤ 0 , we identify the roots as 1, 4, and 5, and test intervals around these roots to find where the expression is non-positive. The solution set is ( − ∞ , 1 ] ∪ [ 4 , 5 ] . This includes the roots since the inequality is non-strict.
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