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Solve the inequality \(\frac{(x - 2)(x - 3)(x - 4)}{(x - 1)(x - 5)(x - 6)} > 0.\)

 Jul 31, 2019
 #1
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\(x\) can't be 2, 3, 4, 1, 5, or 6. Hope this helps! smiley

 Jul 31, 2019
 #2
avatar+105686 
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Certainly x cannot be 1, 5 or 6   becasue that would make the denominaot 0 and you cannot EVER divide by 0.

 

But it will be positive when the top mulutiplies to a positive number

This will happen when 2 factors are negative and the 3rd is positive and ALSO when all three factors are positive

 

So now you have a few scenarios to work out.

 Jul 31, 2019
 #3
avatar+104937 
+1

Note that  

Any x in  (-inf, 1)  will produce  negative / negative  = positive

Any x in (1, 2)  will produce negative / positive  = negative

Any x in (2, 3)  will  produce  pos / pos  = pos  = true

Any x in (3, 4)  will produce neg/ pos  = neg  = false

Any x in (4, 5)  will produce pos/ pos  = pos  = true

Any x in (5, 6) will produce pos/ neg  = neg  = false

Any x in (6, inf) will produce pos/ pos  = pos  = true

 

So.....the intervals that make this true  are  (-inf, 1) U (2,3) U (4,5) U (6, inf)

 

See the graph here : https://www.desmos.com/calculator/ltejhjzirj

 

 

cool cool cool

 Jul 31, 2019
 #4
avatar+105686 
0

Chris I do not think that what you have written here could add to guest's understanding of how to solve this problem.

 

A person who asks this question is not at a level where they would understand the relevance of the graph.

They wouold probably know that you have graphad something in the x z plane but there is no y in the question so they would not understand the relevance.

Melody  Jul 31, 2019
edited by Melody  Jul 31, 2019

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