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My professor wrote this:

$$0<1-\frac{3x}{4}<1$$

$$-1<-\frac{3x}{4}<0$$

$$4>3x>0$$

$$\frac{4}{3}>x>0$$

Is that correct?

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  • $\begingroup$ The first one is closer to be correct, without parenthesis: $0<1-\ 3x/4 <1$. $\endgroup$
    – Berci
    Nov 27 '13 at 0:19
  • $\begingroup$ can you explain why ? and why my is bad ? $\endgroup$ Nov 27 '13 at 0:20
  • $\begingroup$ The answer should go like $-1<-3x<3$ and so $-1<x<1/3$. $\endgroup$ Nov 27 '13 at 0:20
  • $\begingroup$ As Berci said, the professor's workings are right if the question is $0<1-3x/4<1$. As for yours, there are mistakes abound. $\endgroup$ Nov 27 '13 at 0:28
  • $\begingroup$ It is 0<1−(3x/4)<1 With brackets $\endgroup$ Nov 27 '13 at 0:29
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Note: This refers to an earlier version of the question, which has now been substantially changed.

Your professor’s step from

$$0<\frac{1-3x}4<1$$

to $$-1<\frac{-3x}4<0\;,$$

is incorrect: $1$ has been subtracted from $0$ and from $1$, but only $\frac14$ has been subtracted from $\frac{1-3x}4$. Had your professor correctly subtracted $\frac14$ from each term, the result would have been $$-\frac14<-\frac{3x}4<\frac34\;.$$

The next step was to multiply everything by $-4$; doing this to the corrected version leaves you with $$1>3x>-3\;,$$ and dividing everything by $3$ then gives you the result, $$\frac13>x>-1\;.$$

The first step of your calculation is clearly wrong: you’ve multiplied the $1$ by $4$ but not the $\frac{1-3x}4$. Judging by your next step, however, I think that you may have meant to multiply everything by $4$ to get $0<1-3x<4$. Then you wanted to subtract $1$, which is fine, but you must subtract it from the $0$ as well as from the $1-3x$ and the $4$, so you should get $-1<-3x<3$. Dividing everything by $-3$ will now give you

$$\frac13>x>-1\;,$$

as before.

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  • $\begingroup$ it is not (1-3x)/4 it is 1-(3x/4) $\endgroup$ Nov 27 '13 at 0:39
  • $\begingroup$ @user2982390: For the revised question your professor’s work is entirely correct. $\endgroup$ Nov 27 '13 at 0:42
  • $\begingroup$ I don't understand why..or maybe I do, I am very very tired $\endgroup$ Nov 27 '13 at 0:43
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    $\begingroup$ @user2982390: The steps are: $(1)$ subtract $1$ from each of the three expressions; $(2)$ multiply each expression by $-4$, remembering that this reverses the direction of the inequalities, since $-4$ is negative; $(3)$ divide each expression by $3$, which does not change the direction of the inequalities, since $3$ is positive. $\endgroup$ Nov 27 '13 at 0:46

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