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On simple deduction,I know a and b are of opposite sign and b has the positive value. The two numbers can be any arbitrary numbers.

Now in the equation I have to solve,it will be that magnitude of x equals magnitude of ab. Absolute value of ab is always positive,so is absolute value of x,but x can be two numbers->ab,-ab and hence the two answers.

Now it’s clear that ab will lie left of a And right of b. Ex-b=4,a=3.9. Here,ab=-15.6 which lies left of a but in the book the answer is just c and d and none of a and b. How is it possible that two numbers’ product lies between them?

https://i.stack.imgur.com/SrqXL.jpg

Ashish Shukla
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1 Answers1

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if my understanding of the question is correct,here is my idea, if $a,b \in \mathbb{R}$ find $a$ and $b$ s.t. $$a\lt ab\lt b$$.

first, both $a$ and $b$ are nonzero.

from the above inequality, $$\frac{1}{b}\lt 1\lt \frac{1}{a}.$$ if $a\lt 0$ then $$\Rightarrow 1\lt \frac{1}{a}\lt 0$$$$\Rightarrow\!\Leftarrow$$ $$\therefore a\gt 0.$$ case 1: $a\gt 0$ and $b\gt 0$. $$\Rightarrow 1\gt \frac{1}{b}\gt 0\ and\ 0\lt a \lt 1$$ $$\Rightarrow 0\lt a\lt 1\lt b$$.

case 2: $a\gt 0$ and $b\lt 0$. $$b\lt 0\lt a$$$$\Rightarrow\!\Leftarrow$$

so I think the only solution is $$a,b \in \mathbb{R}\ s.t.\ 0\lt a\lt 1\lt b$$ if I'm not wrong.I hope this helps..

euJue07
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