Solve the following inequality. Graph the solution set on a number line.
[tex](x-1)(x-4)(x-5) \leq 0[/tex]
Answer (2)
Find the critical points by setting each factor to zero: x = 1 , 4 , 5 .
Test intervals ( − in f t y , 1 ] , [ 1 , 4 ] , [ 4 , 5 ] , and [ 5 , in f t y ) to determine where the inequality holds.
The inequality holds for ( − in f t y , 1 ] and [ 4 , 5 ] .
The solution set is x in ( − in f t y , 1 ] c u p [ 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 where the expression equals zero. These points are where any of the factors equal zero:
x − 1 = 0 R i g h t a rro w x = 1 x − 4 = 0 R i g h t a rro w x = 4 x − 5 = 0 R i g h t a rro w x = 5
So, the critical points are x = 1 , 4 , 5 .
Creating Intervals Now, we'll use these critical points to divide the number line into intervals and test each interval to see if the inequality ( x − 1 ) ( x − 4 ) ( x − 5 ) ≤ 0 holds true.
The intervals are: ( − in f t y , 1 ] , [ 1 , 4 ] , [ 4 , 5 ] , and [ 5 , in f t y ) .
Testing Intervals Let's test each interval:
( − in f t y , 1 ] : Choose x = 0 . Then ( 0 − 1 ) ( 0 − 4 ) ( 0 − 5 ) = ( − 1 ) ( − 4 ) ( − 5 ) = − 20 ≤ 0 . So, this interval is part of the solution.
[ 1 , 4 ] : Choose x = 2 . Then 0"> ( 2 − 1 ) ( 2 − 4 ) ( 2 − 5 ) = ( 1 ) ( − 2 ) ( − 3 ) = 6 > 0 . So, this interval is not part of the solution.
[ 4 , 5 ] : Choose x = 4.5 . Then ( 4.5 − 1 ) ( 4.5 − 4 ) ( 4.5 − 5 ) = ( 3.5 ) ( 0.5 ) ( − 0.5 ) = − 0.875 ≤ 0 . So, this interval is part of the solution.
[ 5 , in f t y ) : Choose x = 6 . Then 0"> ( 6 − 1 ) ( 6 − 4 ) ( 6 − 5 ) = ( 5 ) ( 2 ) ( 1 ) = 10 > 0 . So, this interval is not part of the solution.
Determining the Solution Set The intervals where the inequality holds true are ( − in f t y , 1 ] and [ 4 , 5 ] . Therefore, the solution set is x in ( − in f t y , 1 ] c u p [ 4 , 5 ] .
Final Answer The solution set is x in ( − in f t y , 1 ] c u p [ 4 , 5 ] . This means x can be any number less than or equal to 1, or any number between 4 and 5, inclusive.
Examples
Understanding inequalities like this helps in various real-world scenarios. For instance, if you're designing a bridge, you need to ensure that the load on the bridge stays within certain limits to prevent it from collapsing. Similarly, in business, you might want to determine the range of prices for a product that will ensure profitability. Inequalities help define these safe or profitable ranges.
To solve the inequality ( x − 1 ) ( x − 4 ) ( x − 5 ) ≤ 0 , we found critical points at 1 , 4 , and 5 and tested intervals to determine where the product is less than or equal to zero. The solution set is x ∈ ( − ∞ , 1 ] ∪ [ 4 , 5 ] . This means the values satisfying the inequality include all numbers less than or equal to 1 and from 4 to 5, inclusive.
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