2
$\begingroup$

If $[x]$ represents the floor of $x$, solve the equation:

$$\left[\frac{x+1}3\right]=\frac{x-1}2$$

Choose the correct answer:

a) $(2,7)\cup(9,15)$

b) $(-3,2)\cup[3,4)\cup(6,14)$

c) $(-1,1]\cup[2,3)\cup(5,8)$

d) $(-5,-3)\cup(1,3]\cup[5,7)$

e) $[1,\frac32]\cup(2,4)\cup[5,7)$

f) $[0,2]\cup[4,7]\cup(9,+\infty)$

Can someone please explain how I can solve this type of equation? Thank you very much!

P.S. I'm not sure if $[x]$ is called whole part of $x$ in English, it's not my native language.

  • 0
    This looks really strange to me, the LHS is always an integer, so the RHS has to be an integer two. Hence $x$ has to be an odd integer. So why are all options intervals?2012-05-24
  • 0
    $$\frac{x-1}{2} \in \mathbb Z$$, so $x$ is an odd number. Seems "e" is the correct answer. Umm, except for $6$!2012-05-24
  • 0
    @Grozav Please comment if you would like more help with some notation that you don't fully understand.2012-05-24

2 Answers 2

4

So as $k:=[\frac{x+1}3] \in \mathbb Z$, we have $\frac{x-1}2 =k \in\mathbb Z$ also. So $x = 2k+1$ is an odd integer. We have to solve $[\frac{2k+2}3] = k$, we now consider three cases.

  • $k = 3\ell$ for some integer $\ell$: Then $$ \left[\frac{2k+2}3\right] = \left[\frac{6\ell+2}3\right] = \left[2\ell + \frac 23\right] $$ Now $2\ell$ is an integer, therefore the integer part of this is $\left[2\ell + \frac 23 \right] = 2\ell$. This equals $k = 3\ell$ for $\ell = 0$, which yields $k=0$, hence $x=1$.

  • $k = 3\ell + 1$ for some $\ell \in \mathbb Z$, here $$\left[\frac{2k+2}3\right] = \left[\frac{6\ell+4}3\right] = \left[2\ell + 1 + \frac 13\right] = 2\ell + 1,$$ which equals $k = 3\ell + 1$ for $\ell = 0$, giving $k = 1$, hence $x = 3$.

  • $k = 3\ell + 2$ for some $\ell \in \mathbb Z$, here $$\left[\frac{2k+2}3\right] = \left[\frac{6\ell+6}3\right] = \left[2\ell + 2\right] = 2\ell + 2,$$ which equals $k = 3\ell + 2$ for $\ell = 0$, giving $k = 2$, hence $x = 5$.

The only solutios are there fore $1$, $3$ and $5$, hence it is (e) (intersected with $2\mathbb Z+1$).

  • 0
    "e" includes $6$ which is not an answer to the equation.2012-05-24
  • 0
    Yes. So I wrote "intersected $2\mathbb Z + 1$". At least (e) doesn't contain wrong odd numbers.2012-05-24
  • 0
    Right, didn't notice that part.2012-05-24
  • 0
    Thank you very much, this is similar to their explanation, but it contains all the steps into getting the results.2012-05-24
  • 0
    There's one thing I didnt exactly understand. When giving k the values of 3l, 3l+1, and 3l+2, why do we equal the floor equation with 2l, 2l+1, 2l+2?2012-05-24
  • 0
    @GrozavAlexIoan I'll edit the above to show more explicit what I was doing, I didn't equate it with $2\ell$, I computed its value. .. Done, HTH.2012-05-24
  • 0
    I understood now. Thank you very much!!!2012-05-24
  • 0
    So none of the answers is correct. What a disappointing question. If this is homework, the teacher does not deserve their job.2012-05-25
1

First of all, notice that $\frac{x-1}{2}$ has to be a whole number (an integer).

Therefore, $x-1$ has to be even, and $x$ has to be an odd integer.

We can use the options to eliminate any number which includes an even number. I guess we are talking only about integers, $\mathbb{Z}$, since any non-integer solutions are definitely out.

For example, the region $(2,7)$ in part (a) includes 4 and 6, so (a) is not correct.

This on its own is not enough to come to a solution, but it should allow us to quickly eliminate wrong options.

In fact, we can go through like this and eliminate all the options that you've given - none of them are right, as it stands.

Perhaps there is a typo in (e)? Since two of the three regions are representing an odd number, but $[5,7)$ incorrectly includes 6. I guess that (e) is supposed to be the correct answer, but maybe it should say $[5,6)$.

  • 0
    You put too much effort into improving your answer, +1.2012-05-24
  • 0
    Thank you for your help, martini's answer contained the practical way into solving the ecuation, however, thank you very much!2012-05-24
  • 0
    @Gigli Haha thanks! Answer first, edit later.2012-05-24
  • 0
    @GrozavAlexIoan You are right to choose that answer, but I think using multiple choice options is an important skill. :)2012-05-24