A fast solution of $\frac{\left|x^2-1\right|-3}{1-2x}<\:x$

Question

If I have this easy inequality

$$\frac{|x^2-1|-3}{1-2x}<\:x$$

your solutions, step by steps are $]-1,\frac{1}{2}[\,\cup\, ]\frac{4}{3},+\infty[$, considering the signs of $|x^2-1|$, i.e. $x^2-1\geq 0 \iff x\leq 1 \vee x\geq 1$ and $x^2-1<0 \iff -1<x<1$ and solving a simple fracture inequality with $1-2x\neq 0$.

I have five options:

\begin{array} {|r|r|}\hline A=]- 1; 1[\,\cup \,] \frac{4}{3} ; 2[ \\ \hline B=] -\infty;-1[ \cup ] \frac{1}{2} ; \frac{4}{3} [ \\ \hline C=]-1;\frac{1}{2}[ \cup ] ; ]\frac{4}{3} +\infty[ \\ \hline D=] -\infty;-1[ \cup [1; \frac{4}{3} [ \\ \hline \text{None of the previous answers are correct}\\ \hline \end{array} I've seen, without to do the calculus, that the $0$ satisfies the inequality and it is not are into the sets $B$ and $D$. It not $A$ because $x\neq \frac 12$. Hence I have $50 \%$ to find the correct answer: or it is into $C$ (exact answer) or $E$ (none of the previous answers are correct).

Any of you users see something else in your minds?

I think you mistyped the last sentence: you presumably didn't want to say D but E (previously having labelled the last answer by E). I can see people are trying to prove to you that D is not a solution - which you have already spotted! — , Aug 23 '20 at 16:41
@StinkingBishop Ops...I have done a mistake....thankkkkkk youuu. Now I edit my question thank you again very much. — Sebastiano, Aug 23 '20 at 19:02

mathcounterexamples.net · Accepted Answer · 2020-08-24T07:34:52.327

2

It can’t be D as

$$\lim\limits_{x \to -\infty} \frac{|x^2-1|-3}{1-2x}= \infty$$

Therefore for large negative values of $x$, $\frac{|x^2-1|-3}{1-2x}$ will be positive and the requested inequality can’t be satisfied. This allows to drop down option D that contains the interval $(-\infty,-1)$.

edited Aug 24 '20 at 07:34

answered Aug 23 '20 at 16:11

mathcounterexamples.net

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Why do you consider the $\lim$? Is there a true valid reason? Can you, please, explain better your answer. In meantime +1. – Sebastiano Aug 23 '20 at 16:15
I consider limits because it is easy to evaluate for fraction maps. – mathcounterexamples.net Aug 23 '20 at 16:18
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What do you mean? This methodology never done in my life. Could you please add a few more specific details so that I can improve my knowledge? – Sebastiano Aug 23 '20 at 16:23
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What do I need to clarify better? The computation of the limit (which requires some knowledge on rational fractions)? Or why the value $\infty$ for the limit allows to drop down option D? – mathcounterexamples.net Aug 23 '20 at 18:37
Because the value $∞$ for the limit allows to drop down option $D$ comparated with all the inequality: $\frac{|x^2-1|-3}{1-2x}<:x$.What is essentially the usefulness of the limit? Obviously if I study the function at the first member I observe if it respects the inequality with respect to the graph $g(x)=x$. – Sebastiano Aug 23 '20 at 19:08
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See update in my answer. – mathcounterexamples.net Aug 24 '20 at 06:24
Approved and thank you very much with kindness. – Sebastiano Aug 24 '20 at 18:52

score 1 · Answer 2 · answered Aug 23 '20 at 16:49

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On one hand, the perfectionist in me shudders. On the other hand, you have reduced the probability that your "random" guess would be wrong from $80\%$ to $50\%$ (which is not by a small amount - well done!), all by using a few simple but sound mathematical arguments. It is a bit of gaming the system, but unless teachers abandon using the multiple-choice-question tests in favour of old-fashioned questions ("Find the set of solutions of the inequality ... Show your reasoning.") which are way harder to mark, the system is there to be gamed, I suppose.

answered Aug 23 '20 at 16:49

As written in my last questions I am helping a university student of Economics and Finance who does not have the appropriate knowledge. I wonder what some of the questions included in his tests will be used for in his field. – Sebastiano Aug 23 '20 at 19:12
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@Sebastiano I see where you are coming from. I think it is great that you have shown the student that some problems can be efficiently solved with right ideas, rather than blindly following the rules. ("Thinking outside the box"). I do think that the student should also know how to solve the same problem "the long way", as (1) Sometimes the right idea just does not come to mind, and (2) Doing things in a methodical and pedantic way is a useful skill (attitude, anyways) in itself, and likely useful for the student's future job, where omissions will cost real money. My 2p. – Aug 23 '20 at 19:25

A fast solution of $\frac{\left|x^2-1\right|-3}{1-2x}<\:x$

2 Answers2