Is ((p - 3)^2)^(1/2) = 3 - p ? (1) p p

Question

Is  \sqrt{(p - 3)^2} = 3 - p  ?

(1)  p < |p|

(2)  3 > p

chetan2u · Answer

Let us take p-3 as x for ease of understanding The square root or for that matter any even root will always be non negative. So, \sqrt{x^2}=|x| Now since square root cannot be negative result, \sqrt{x^2}\geq{0} . The square root therefore would also depend on whether x is negative or positive. 1) x\geq{0} => \sqrt{x^2}=x 2) x<0 => \sqrt{x^2}=-x The above is for understanding, so let us get back to the question If p-3 =x, then 3-p=-x Thus we are asking whether \sqrt{x^2}=-x , and this happens when x is negative. So we are asking whether x or p-3 is negative => p-3<0.....p<3? (1) p < |p| p=|p| if p is positive but p will be less than its absolute value only when p<0. We are looking for whether p<3. Answer is yes as p<0<3 Sufficient (2) 3 > p This is exactly what we were looking for. Sufficient D

mx39 · Answer

While this one is heavy on the square roots, I think the wording of the statements means this is truly a test of number properties. Because our statements attach constants to the variables, I knew testing for numbers would likely be an efficient approach. That said, here's how I tackled this.

Step One: Simplify (don't FOIL) the Stimulus
If we're asked whether \(\sqrt{(p-3)^2} = 3 - p\), we can remove the radical and re-express it as: \((p-3)^2 = (3-p)^2\)?

Step Two: Check Against Statements
As I mentioned above, I really don't think multiplying out those numbers helps us at all in determining equality. The question is presented in factored form, so it'll be easy to check against numbers now.

(1) \(p < |p|\)

In plain English, statement one is telling us p is negative. That's it. So if we test a handful of negative numbers, let's see if we get a consistent answer. I like to test really widely spaced out numbers to make sure we're not falling into any traps. Let's try -1, -10, and -25.

-1: Is \((-1 - 3)^2 = (3 - (-1))^2\)? ==> Is \(-4^2 = 4^2\)? Yes!
-10: Is \((-10 - 3)^2 = (3 - (-10))^2\)? ==> Is \(-13^2 = 13^2\)? Yes!
-25: Is \((-25 - 3)^2 = (3 - (-25))^2\)? ==> Is \(-28^2 = 28^2\)? Yes!

Statement One alone is sufficient. Eliminate B, C, and E.

(2) p < 3

Here, we can plug in numbers intelligently. We know we get a consistent answer with negative values, so let's try for 0, 1, and 2.

0: Is \((0 - 3)^2 = (3 - 0)^2\)? ==> Is \(-3^2 = 3^2\)? Yes!
1: Is \((1 - 3)^2 = (3 - 1)^2\)? ==> Is \(-2^2 = 2^2\)? Yes!

By now we're starting to notice a pattern. If you need extra reassurance, continue for 2.

2: Is \((2 - 3)^2 = (3 - 2)^2\)? ==> Is \(-1^2 = 1^2\)? Yes!

Statement Two alone is sufficient. Eliminate A.

AC D.

Is ((p - 3)^2)^(1/2) = 3 - p ? (1) p < |p| (2) 3 > p

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