Inequalities - 1

[bubbliiiiiiii]

If p is an integer greater than zero, is 200 < sqrt(p)?

198 < sqrt(p-2)
202 < sqrt(p+2)

[spoiler]OA: B[/spoiler]

Looking for explanations and alternative approaches.

[HSPA]

198^2+2 cannot reach 200^2...

[manpsingh87]

If p is an integer greater than zero, is 200 < sqrt(p)?

198 < sqrt(p-2)
202 < sqrt(p+2)

[spoiler]OA: B[/spoiler]

Looking for explanations and alternative approaches.

1) 198<sqrt(p-2); when p= 40002, then p-2=40002-2=40000 therefore sqrt(40000)=200
198<200, also 200<sqrt(40002);

now consider p=199^2, then p-2=199^2-2 will be greater than 198 but 200<199 is not true,
as here different answers are possible for different values of p hence a alone is not sufficient to answer the question.

2)202<sqrt(p+2); now the minimum value of p=202^2, for which this inequality will be true,
also 200<202, is true, therefore 2 alone is sufficient to answer the question hence B..!!

[Ian Stewart]

If p is an integer greater than zero, is 200 < sqrt(p)?

198 < sqrt(p-2)
202 < sqrt(p+2)

[spoiler]OA: B[/spoiler]

Looking for explanations and alternative approaches.

I'd get rid of the roots first, since they're likely to be confusing (we can square both sides of an inequality if we know both sides are positive, which is the case here). The question asks if p > 200^2.

Statement 1 tells us that p-2 > 198^2, or that p > 198^2 + 2. Now, 198^2 + 2 is only slightly greater than 198^2, so is certainly less than 200^2, so we don't know whether p is greater than 200^2.

Statement 2 tells us that p+2 > 202^2, or that p > 202^2 - 2. Now 202^2 - 2 is only slightly less than 202^2, so is certainly more than 200^2, and Statement 2 is sufficient.

In evaluating each statement, I've made use of the fact that the distance between 198^2 and 200^2 is greater than 2, and the distance between 200^2 and 202^2 is greater than 2. If you need to, you can use the difference of squares to work out just how far apart those numbers actually are: 200^2 - 198^2 = (200 + 198)(200 - 198) = 398*2 = 796. You can do the same for 202^2 - 200^2.