Is the product of a positive and a negative integer less than -10?

You forgot that your numbers are integers. Therefore, \(x\geq{4}\) and \(y\leq-3\). From here, continue as you did above.
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So lets say the positive integer is 'x' and the negative integer is 'y'. We have to determine if x*y < -10

(1) x > 3. So the minimum value of x is 4 (has to be an integer). But nothing about y, so insufficient.

(2) y < -2. So maximum value of y is -3 (has to be an integer). But nothing about x, so insufficient.

Combining the two statements, if we take x = 4 and y = -3, then their product = 4*-3 = -12. For any other values of x & y satisfying the two statements, product will be less than -12 only. (eg, x=5, y=-3 gives us -15 and x=4, y=-4 gives us -16 and so on...)
So if product is less than -12, it will obviously be less than -10 also. (any number to the left of -12 will also be to the left of -10 on the number line). Sufficient.

Hence C answer

Is the product of a positive integer and a negative integer less than -10?

(1) The positive integer is greater than 3.

(2) The negative integer is less than -2.

Yes consider +ve no 4 and -ve no -3 multiplication gives -12 which is less than -10.

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Merging topics. Please check the discussion above.
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Positive integer p > 3 = 4,5,6,....
Negative integer n < -2 = -3,-4,-5...

pn = 4*-3 = -12 <-10
other products of p*n will be less than -12 since magnitude of p & n will increase.

IMO C
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Be careful with wording. One positive integer and one negative. Great. is product < -10?

We can see each individual is not sufficient.

Combined:

Positive integer is at least 4 and negative integer is at least -3. So the product is at least -12. Answer = Yes.

Final Answer C
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To answer this question, we need the solid understanding that as the magnitude of negative numbers increases, their value decreases. so, in the negative number universe, -3>-10.

With this in mind, we are ready to take on this question. So let us begin.

S1. take x=-ve and y=+ve. We can take y=4 as per this statement since the only constraint is that y>3. We should actively try and get opposing answers to 'yes/no' kind of questions. Let's do yes first.
so we established that y=4; let us assume x=-10 (as it can be any negative number). x*y=4*-10=-40 <-10. Remember the rule negative nu,ber rule above.
Now let's see of we can get a 'no' and call it a day: y is till 4, let us take x=-1 (remember, we are purposely trying to prove no here)
xy=4*-1=-4 which is NOT less than -10. Hence disproved and this has rendered the statement insufficient. Strike A and D off.

S2. Take x=-3 (-3<-2). again let's purposely try and get a yes and no.
Yes first: y= 10 (since there are no constraints on y). xy= -3*10= -30<-10. ok.
No: y= 1: xy= -3*1= -3 which is NOT <-10. Hence disproved and this has rendered the statement insufficient. Strike B off as well.

Now combining the two, we can solve for 3 or 4 cases and we will notice that all of them answer in 'Yes' therefore the answer here is C.
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