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Question: Question

My solution: My solution

According to me -6 is least possible integer which satisfies option d.

Textbook's solution Books solution

I don't understand what I'm doing wrong. Thank you for your help.

user3508140
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  • There is no way the solution can be -7, if you put -7 in the inequality, you will get a $\infty$ term in the denominator. Your $-7$ is in an open interval, not a closed one. The interval you're looking for is $(-7, 2)$ so your numbers can be ${-6,-5,-4...1}$. This makes $-6$ the next option and I'm not exactly sure why your TextBook has not mentioned $6$ as the right answer.... –  Oct 05 '18 at 13:53
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    Right. -7 can't be the answer. I had put an open bracket in my solution, but made a mistake while framing the question. I'll edit it out. – user3508140 Oct 05 '18 at 13:59

2 Answers2

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So we are given the inequality:

$$ \frac{\left(x-5\right)}{\left(x+7\right)\left(x-2\right)}>0 $$

Applying wavy-curve like you did or just looking at the graph of the inequality, we get our solution interval as:

$$ x\in (-7,2) \cup (5, \infty) $$

I have verified this answer over WolframAlpha:

enter image description here

Hence the least integral value that $x$ can take would be $-6$, followed by $-5,-4,-3...1$

I think we can see that $-6$ satisfies option D as :

$$ f(x) = x^2+5x-6, \ \ f(-6) = 0$$

So I am guessing that the correct answer should be $-6$.

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Multiplying both sides by the square of the denominator on LHS gives $$(x-5)(x^2+5x-14)>0,$$ which factors as $$(x-5)(x+7)(x+2)>0.$$

Thus the values of $x$ satisfying the inequality are $-7<x<2$ or $x>5.$

So we can see that the least integer satisfying the inequality is $-6.$

Allawonder
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