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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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    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 $12y \lt 8x$.2012-07-09

2 Answers 2

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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.