I am learning to work with inequalities. For an example, let's say I want to find the set of $(x, y) \in \mathbb{R}^{2}$ that satisfy: $|x \cdot y| \leq 2$.
Determine and sketch the pairs $(x, y)$ in $\mathbb{R} \times \mathbb{R}$ that satisfy: $|x\cdot y| \leq 2$
3 Answers
HINT: When you look at the answer of Chandru1 you must think twice. Do you know any algebraic properties of the absolute value? How do you solve an inequality?
You might also like to think of different points $(x,y)=(5,5)$, $(x,y)=(5,-5)$, $(x,y)=(-5,5)$, $(x,y)=(-5,-5)$ -- are they in you set? $(x,y)=(1/5,5)$, $(x,y)=(1/5,-5)$, $(x,y)=(-1/5,5)$, $(x,y)=(-1/5,-5)$ -- are they in you set?
My suggestion:
(A) Start with the definition of absolute value $\left\vert z\right\vert $, with $z=xy,x,y\in\mathbb{R}$
$$\left\vert xy\right\vert =xy\qquad \text{if}\ xy\geq 0\iff \left( x\geq 0\wedge y\geq 0\right) \vee \left( x\leq 0\wedge y\leq 0\right) $$
$$\left\vert xy\right\vert =-xy\qquad \text{if}\ xy<0\iff \left( x\geq 0\wedge y\leq 0\right) \vee \left( x\leq 0\wedge y\geq 0\right) $$
(B) Add the condition $\left\vert xy\right\vert \leq 2\iff -2\le xy\leq 2$.
(Why?)
(C) Observe that
(i) $xy\gt 0$
in the 1st and 3rd quadrants; and
(ii) $xy\lt 0$
in the 2nd and 4th quadrants.
(Why?)
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0I edited the format, breaking the lines, due to incorrect display of text mixed with symbols. – 2010-10-16
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0Thank you for correcting the format. – 2010-10-16
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0@Justin: You are welcome! The changes I made was in the format of my answer, not in your question. As far as I can see one is allowed to change others posts only when his/her rep is at least 1000. – 2010-10-16
By taking the seperate cases where $x = 0$ or $y = 0$ and showing that they are in the set, then we can assume $x,y \ne 0$ Therefore we can take $-2 \le xy \le 2$.
Therefore:
- $-2/y \le x \le 2/y$
- $-2/x \le y \le 2/x$
So for all x,y real numbers satisfying the above, and (0, y real number) (x real number, 0), x,y are in R x R.
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0ftp://ftp.ams.org/pub/tex/.../amsmath/short-math-guide.pdf is the most concise one I know, you basically just need to know that dollar signs start math commands and peruse the pdf for whatever symbol you need. http://www.codecogs.com/latex/eqneditor.php also allows you to do this online, if you find that easier – 2010-10-16
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0I need to cleen this up to take into account positive and negative x and y. Other than that it should be ok. – 2010-10-20