# Issue with combining InEqualities

I have the following two inequalities: \begin{align*} 8x &\gt 12y\\ 12y &\gt 15z \end{align*} Now the book states that we need to line up the inequalities as such $$\begin{array}{rcccccl} 0 &<& 15z\\ && 15z &<& 12y\\ & & & & 12y &<& 8x\\ \end{array}$$ Hence we get $$0 < 15z < 12y < 8x$$

Now my question is how did the book get the following $$\begin{array}{rcccccl} 0 &<& 15z\\ && 15z &<& 12y\\ && && 12y &<& 8x \end{array}$$

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Thats what i want to know how did the book get $$0<15z$$ , I could only come up with $$15z < 12y < 8x$$ –  Rajeshwar Jul 9 '12 at 23:05
The $0\lt 15z$ part is not sensible, since nothing in the two given inequalities implies that $0 \lt 15z$. As to the others, $p \gt q$ says the same thing as $q \lt p$. So for example the given inequality $8x \gt 12y$ can be rewritten as $12y \lt 8x$. –  André Nicolas Jul 9 '12 at 23:06

You know that $12y\lt 8x$, because that is the first inequality you have; it appears last in the large display.

You also know that $15z\lt 12y$, because that is the second inequality you have; it appears in the middle of the large display.

And, presumably, you know that $z$ is positive, so that $15z$ is also positive, $15z\gt 0$.

So: $0$ is smaller than $15z$; and $15z$ is smaller than $12y$; and $12y$ is smaller than $8x$. That's the three inequalities that appear, only they are indented to make it clear how they fit together.

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When you write a compound inequality such as $1 < x < 2$, you are using a tacit Boolean and. So, for example, $$(1 < x < 2) \Leftrightarrow ((1 < x) \wedge (x < 2)).$$ This is how you should parse it.

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