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

Guest Jul 31, 2019

#2**+1 **

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.

Melody Jul 31, 2019

#3**+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

CPhill Jul 31, 2019

#4**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