Solve the following inequality. Graph the solution set on a number line.

[tex](x-1)(x-4)(x-5) \leq 0[/tex]

Find the critical points by setting each factor to zero: x = 1 , x = 4 , and x = 5 .
Create intervals using the critical points: ( − ∞ , 1 ] , [ 1 , 4 ] , [ 4 , 5 ] , and [ 5 , ∞ ) .
Test a value from each interval in the inequality ( x − 1 ) ( x − 4 ) ( x − 5 ) ≤ 0 to determine if the interval satisfies the inequality.
The solution set is ( − ∞ , 1 ] ∪ [ 4 , 5 ] .

Explanation

Understanding the Inequality We are given the inequality ( x − 1 ) ( x − 4 ) ( x − 5 ) ≤ 0 . Our goal is to find all values of x that satisfy this inequality. This involves finding the intervals where the product of the three factors is either negative or zero.

Finding Critical Points First, we need to find the critical points, which are the values of x that make each factor equal to zero. These points are where the expression can change its sign. We set each factor to zero and solve for x :


x − 1 = 0 ⟹ x = 1
x − 4 = 0 ⟹ x = 4
x − 5 = 0 ⟹ x = 5
So, the critical points are x = 1 , x = 4 , and x = 5 .

Creating Intervals Now, we use these critical points to divide the number line into intervals. The intervals are ( − ∞ , 1 ] , [ 1 , 4 ] , [ 4 , 5 ] , and [ 5 , ∞ ) . We will test a value from each interval to see if the inequality ( x − 1 ) ( x − 4 ) ( x − 5 ) ≤ 0 holds true.

Testing Intervals Let's test each interval:

Interval ( − ∞ , 1 ] : Choose x = 0 . Then, ( 0 − 1 ) ( 0 − 4 ) ( 0 − 5 ) = ( − 1 ) ( − 4 ) ( − 5 ) = − 20 . Since − 20 ≤ 0 , this interval satisfies the inequality.

Interval [ 1 , 4 ] : Choose x = 2 . Then, ( 2 − 1 ) ( 2 − 4 ) ( 2 − 5 ) = ( 1 ) ( − 2 ) ( − 3 ) = 6 . Since 0"> 6 > 0 , this interval does not satisfy the inequality.

Interval [ 4 , 5 ] : Choose x = 4.5 . Then, ( 4.5 − 1 ) ( 4.5 − 4 ) ( 4.5 − 5 ) = ( 3.5 ) ( 0.5 ) ( − 0.5 ) = − 0.875 . Since − 0.875 ≤ 0 , this interval satisfies the inequality.

Interval [ 5 , ∞ ) : Choose x = 6 . Then, ( 6 − 1 ) ( 6 − 4 ) ( 6 − 5 ) = ( 5 ) ( 2 ) ( 1 ) = 10 . Since 0"> 10 > 0 , this interval does not satisfy the inequality.

Determining the Solution Set The intervals that satisfy the inequality are ( − ∞ , 1 ] and [ 4 , 5 ] . Therefore, the solution set is the union of these intervals: ( − ∞ , 1 ] ∪ [ 4 , 5 ] .

Graphing the Solution Set To graph the solution set on a number line, we shade the intervals ( − ∞ , 1 ] and [ 4 , 5 ] . We use closed circles at x = 1 , x = 4 , and x = 5 to indicate that these points are included in the solution set.


Examples
Understanding inequalities like this is crucial in many real-world applications. For example, in engineering, you might use inequalities to determine the range of acceptable values for a component's performance. If the performance falls outside this range, the component might fail. Similarly, in economics, inequalities can help model constraints on resources or production levels. For instance, a company might need to ensure that its production costs stay below a certain threshold to remain profitable. By solving inequalities, businesses can make informed decisions about pricing, production, and resource allocation, ensuring they operate within acceptable limits and maximize their success.

Answered by GinnyAnswer | 2025-07-08

To solve the inequality ( x − 1 ) ( x − 4 ) ( x − 5 ) ≤ 0 , we find the critical points at x = 1 , x = 4 , and x = 5 , create intervals, and test them to determine the solution set, which is ( − ∞ , 1 ] ∪ [ 4 , 5 ] . We graph this solution on a number line using closed circles at the critical points to show they are included in the solution.
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Answered by Anonymous | 2025-08-25