Lesson: Strange Operators

Let's take it from (a+1)/(b-1) = 1
Multiply both sides of the equation by (b-1) to get: (a+1) = 1(b-1)
Expand right side: a + 1 = b - 1
Subtract 1 from both sides: a = b - 2

Answer: A

Question link: https://gmatclub.com/forum/if-x-y-x-y-x-y-then-218476.html

You made a small mistake in your solution.

You're using the property that says a² - b² = (a + b)(a - b)

So, in the expression, (√5 + √5)² - (√5 - √5)², what is a, and what is b?
Well, a = (√5 + √5) and b = (√5 - √5)

This means (a + b)(a - b) = [(√5 + √5) + (√5 - √5)][(√5 + √5) - (√5 - √5)]
= [(√5 + √5) + 0][(√5 + √5) - 0]
= (√5 + √5)(√5 + √5)
= (2√5)(2√5)
= 4√25
= 20

I have two different solutions posted here: https://gmatclub.com/forum/if-x-y-x-y-x-y-then-218476.html#p1727422

Does that help?

Cheers,
Brent

Solution link: https://gmatclub.com/forum/for-all-positive-integers-m-3m-when-m-is-odd-...

Great point!
[162] would also be correct.

Cheers,
Brent

Hi Swatato,

Here's the solution you're referring to: https://gmatclub.com/forum/for-any-real-number-x-the-operator-is-defined...

I spent the first part of the solution, showing that &(p + 1) = (p + 1)(-p) [upon simplification]
ASIDE: This step is crucial, because once we know that &(p + 1) and (p + 1)(-p) are EQUAL, we can later replace &(p + 1) with (p + 1)(-p).

Then I dealt with the given information that says: p + 1 = &(p + 1)

So, I took the part on the right side of the equation, &(p + 1), and replaced it with its equivalent value of (p + 1)(-p)

This results in the equation p + 1 = (p + 1)(-p), which we can now solve for p.

Does that help?

Cheers,
Brent

Question link: https://gmatclub.com/forum/an-operation-is-defined-by-the-equation-x-y-x...

You're correct to say that x²/4 - xy - y² = (x/2 - y)².
However, I think it's safe to say that most students will not automatically see that factorization.
So, as an intermediate step, I factored out the 1/4 so that the resulting quadratic is easier to recognize as a special product.

Cheers,
Brent

Good catch!

You're partially right about how I should have rephrased the target question.

Notice that:
[2.1] = 3, since 3 is the smallest integer that's greater than or equal to 2.1.
[0] = 0, since 0 is the smallest integer that's greater than OR EQUAL TO 0
[1] = 1, since 1 is the smallest integer that's greater than or equal to 1.

The target question asks whether [x] = 0
This will occur if -1 < x ≤ 0

So, the REPHRASED target question should be "Is -1 < x ≤ 0"

I've edited my answer here: https://gmatclub.com/forum/if-denotes-the-least-integer-greater-than-or-...

Thanks again for the heads up!

Cheers,
Brent

GIVEN: &(x) = x(1 − x)

So, the "&" symbol next to a number tells us to take that number and multiply it by that number subtracted from 1.

In other words, &number = number(1 - number)
So, &7 = 7(1 - 7)
And &88.3 = 88.3(1 - 88.3)
And &w = w(1 - w)
And &2k = 2k(1 - 2k)
And &j² = j²(1 - j²)
And &(p+1) = (p+1)[1 − (p+1)]

In all cases, we're taking the number (be it 7 or 88.3 or w or 2k or j²) and multiplying it by that number subtracted from 1.

Does that help?

Cheers.
Brent

The first step is to figure out what the strange notation means.

GIVEN: [y] denotes the greatest integer less than or equal to y
A few examples:
[5.1] = 5
[3] = 3
[8.9] = 8
[-1.4] = -2
[-13.6] = -14

Once we have a better understanding of what the notation means, we can answer the question.
Here's my full solution: https://gmatclub.com/forum/denotes-the-greatest-integer-less-than-or-equ...

Please let me know if you have any questions about my solution.

Cheers,
Brent