Find the values of a real number ##a## for an inequality to hold

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The discussion revolves around finding the values of a real number a such that the inequality (x - (a-1))(x - (a^2 + 2)) < 0 holds for all x in the interval (-1, 3). The initial attempt yielded values a = 0, ±1, but these were shown not to satisfy the conditions of the problem. The book's solution suggests a ≤ -1, which was initially questioned but later confirmed as correct, as it allows the inequality to hold in a larger interval that includes (-1, 3). Ultimately, it was clarified that no values of a can satisfy both constraints simultaneously, leading to the conclusion that the permissible set of values for a is empty.
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Homework Statement
Find the set of all possible real values of ##a## such that the inequality ##(x-(a-1))(x-(a^2+2))<0## holds for all ##x \in (-1,3)##
Relevant Equations
If ##\boldsymbol{(x-a)(x-b)<0\Rightarrow x \in (a,b)}##, where ##a<b##. We remember that the "interval" ##(a,b)## stands for the set of all real numbers ##x## such that ##a < x < b##.
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Problem statement :
Let me copy and paste the problem as it appears in the text.
Attempt : From the "Relevant Equations" given above, we can compare to see that ##a-1 = -1## and ##a^2+2=3##. These lead (after some algebra) to the three values of ##\boxed{a=0, \pm 1}##.

Issue : The book has a different answer. It says ##\boxed{a\le -1}##.

Book's solution : I copy and paste the solution given in the book below.

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Doubt : Besides not following the solution in the text, let's investigate its answer to the problem, viz. ##\boxed{a\le -1}##. So let me take a value of ##a##, say ##a=-3##. That would make the inequality read ##(x+4)(x-11)<0##. This would lead to the following interval for ##x \in (-4,11)##, clearly different from the given interval [##x \in (-1,3)##].

Is the book mistaken? A hint or a suggestion would be welcome.
 
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brotherbobby said:
Homework Statement:: Find the set of all possible real values of ##a## such that the inequality ##(x-(a-1))(x-(a^2+2))<0## holds for all ##x \in (-1,3)##

Attempt : From the "Relevant Equations" given above, we can compare to see that ##a-1 = -1## and ##a^2+2=3##. These lead (after some algebra) to the three values of ##\boxed{a=0, \pm 1}##.

If we take ##a = 0## (one of your solutions), then we need:
$$\forall x \in (-1, 3): (x+1)(x-2) < 0 $$This fails for ##x = 2.5##, say. So ##a=0## is not a solution.

Likewise for ##a = 1## we need:
$$\forall x \in (-1, 3): (x)(x-3) < 0 $$This fails for ##x = 0##, say. So ##a=1## is not a solution.

For ##a = -1## we need:
$$\forall x \in (-1, 3): (x+2)(x-3) < 0 $$The inequality holds on that interval. So ##a=-1## is a solution.
brotherbobby said:
Doubt : Besides not following the solution in the text, let's investigate its answer to the problem, viz. ##\boxed{a\le -1}##. So let me take a value of ##a##, say ##a=-3##. That would make the inequality read ##(x+4)(x-11)<0##. This would lead to the following interval for ##x \in (-4,11)##, clearly different from the given interval [##x \in (-1,3)##].


The interval ##(-4, 11)## contains the interval ##(-1, 3)##, so the inequality holds on this interval. As required. The question does not require that the inequality holds only on the specified interval.
 
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You cannot require both a -1 = -1 and a^2 + 2 = 3: The first requires a = 0, the second requires a = \pm 1. You do not get three possible values for a from these conditions; you get zero possible values of a.

So in fact there is no choice of a for which (x - (a-1))(x - (a^2 + 2)) &lt; 0 holds only for x \in (-1,3) and not otherwise; the book's solution that a must satisfy both a - 1 \leq -1 and a^2 + 2 \geq 3 and the inequality hold for some larger interval (which cannot be made smaller than (-2,3) when a = -1) is correct.
 
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pasmith said:
You cannot require both a -1 = -1 and a^2 + 2 = 3: The first requires a = 0, the second requires a = \pm 1. You do not get three possible values for a from these conditions; you get zero possible values of a.

So in fact there is no choice of a for which (x - (a-1))(x - (a^2 + 2)) &lt; 0 holds only for x \in (-1,3) and not otherwise; the book's solution that a must satisfy both a - 1 \leq -1 and a^2 + 2 \geq 3 and the inequality hold for some larger interval (which cannot be made smaller than (-2,3) when a = -1) is correct.
It's with the first paragraph of your response that I am struggling with. If a variable like ##a## can have values less than or equal to 1 (##a\le 1##) which is the answer, surely ##a## can take a range of values and yet satisfy the given problem situation. Why then can it not be 0 and ##\pm 1##? Surely it could have taken those three values. It doesn't, for as @PeroK has shown in post #2, they would violate the problem conditions. My question is that there is nothing intrinsically wrong with a variable having two or more values, is there?
 
PeroK said:
The question does not require that the inequality holds only on the specified interval.
Thank you @PeroK. I suppose this is the main place where I was mistaken. The inequality would hold in an interval beyond the one that is given, viz. ##(-1,3)##.
 
brotherbobby said:
Thank you @PeroK. I suppose this is the main place where I was mistaken. The inequality would hold in an interval beyond the one that is given, viz. ##(-1,3)##.
Suppose you had a train ticket and wanted to travel on a Wednesday. So, you ask whether the ticket is valid on Wednesday? Suppose the ticket is valid Monday to Friday. What answer would you expect:

A) No. The ticket is not valid on Wednesday- it's valid Monday to Friday.

B) Yes. The ticket is valid on Wednesday- because Wednesday is a subset of Monday to Friday.

From my point of view, "no" would be a strange, illogical answer
 
PeroK said:
Suppose you had a train ticket and wanted to travel on a Wednesday. So, you ask whether the ticket is valid on Wednesday? Suppose the ticket is valid Monday to Friday. What answer would you expect:

A) No. The ticket is not valid on Wednesday- it's valid Monday to Friday.

B) Yes. The ticket is valid on Wednesday- because Wednesday is a subset of Monday to Friday.

From my point of view, "no" would be a strange, illogical answer
Yes, it would be B), in answer to your question.

What made my given problem above in post# 1 a bit tricky is that there was apparently "two" intervals, one in the variable ##a## [##(a-1, a^2+2)##] and another in integers [##(-1,3)##]. A student might equate them, as I did. Doing so however gave wrong answers for the value of ##a##. The trick is to realize that the interval given by variable ##a## includes that given with the integers.
 
brotherbobby said:
It's with the first paragraph of your response that I am struggling with. If a variable like ##a## can have values less than or equal to 1 (##a\le 1##) which is the answer, surely ##a## can take a range of values and yet satisfy the given problem situation. Why then can it not be 0 and ##\pm 1##?

a must lie in a set of permissible values. When you impose the constraint a -1= -1 you limit that set to just \{0\}. None of the members of this set satisfy the further constraint a^2 +2= 3. That \pm 1 do satisfy this constraint does not assist you: the first constraint already excluded them from consideration. The set of permissible values is empty.

It is in this sense that you cannot impose both constraints: they do not lead to a solution.
 
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