Is sqrt((y-4)^2) = 4 - y?

[BekermanLeo]

Is sqrt((y-4)^2) = 4 - y?

1) |y-3| <= 1
2) y * |y| > 0

Source: MGMAT Advanced Quant Q76 on Workout Set 8.

I was hoping I could get an explanation for this question - I looked on the back and I don't understand why 'Is -(y-4) >= 0?' is a rephrase of '|y-4| = 4 - y'. Could I get some help with this? Thank you so much.

[RonPurewal]

First you want to make sure that you understand the basics of what the square/square root combination is doing.

Consider just the expression √(x^2); forget the (y - 4) for now.
* If you plug a positive number in for x, you'll get the same number back out. (Try plugging in a few numbers if you don't already see this.)
* If you plug a negative number in for x, you'll get the opposite number (i.e., made positive) back out.
In other words, √(x^2) is the same as ...
... x itself, if x is positive;
... -x (which is a positive number!), if x is negative.
If x = 0, then √(x^2) = x and √(x^2) = -x are both true statements.

So there's the relationship.

More generally, this means ...
... √(stuff^2) = stuff, if the "stuff" is positive;
... √(stuff^2) = opposite of the stuff, if the "stuff" is negative.
If the stuff is 0, then both of these are true.

Taking that last case into account, the most general form of this fact is:
* √(stuff^2) = stuff --> means the stuff ≥ 0.
* √(stuff^2) = opposite of the stuff --> means the stuff ≤ 0.

If "the stuff" is y - 4, then 4 - y is the opposite of that stuff.
So...
* √((y - 4)^2) = y - 4 means y - 4 ≥ 0, or, in other words, y ≥ 4.
* √((y - 4)^2) = 4 - y means y - 4 ≤ 0, or, in other words, y ≤ 4.

Hope that helps.

[hkparikh09]

Is the answer "E"?

[jnelson0612]

Is the answer "E"?

The answer is A.

[mafcostanza]

First you want to make sure that you understand the basics of what the square/square root combination is doing.

Consider just the expression √(x^2); forget the (y - 4) for now.
* If you plug a positive number in for x, you'll get the same number back out. (Try plugging in a few numbers if you don't already see this.)
* If you plug a negative number in for x, you'll get the opposite number (i.e., made positive) back out.
In other words, √(x^2) is the same as ...
... x itself, if x is positive;
... -x (which is a positive number!), if x is negative.
If x = 0, then √(x^2) = x and √(x^2) = -x are both true statements.

So there's the relationship.

More generally, this means ...
... √(stuff^2) = stuff, if the "stuff" is positive;
... √(stuff^2) = opposite of the stuff, if the "stuff" is negative.
If the stuff is 0, then both of these are true.

Taking that last case into account, the most general form of this fact is:
* √(stuff^2) = stuff --> means the stuff ≥ 0.
* √(stuff^2) = opposite of the stuff --> means the stuff ≤ 0.

If "the stuff" is y - 4, then 4 - y is the opposite of that stuff.
So...
* √((y - 4)^2) = y - 4 means y - 4 ≥ 0, or, in other words, y ≥ 4.
* √((y - 4)^2) = 4 - y means y - 4 ≤ 0, or, in other words, y ≤ 4.

Hope that helps.

Ron, thank you for the explanation, this is beginning to make sense to me, you have an amazing ability to explain tough problems in a digestible way.
I just have one question or issue I hope you can explain/relate to your explanation above:

In chapter 2 of the Algebra workbook we are taught how to solve Absolute Value Equations by "considering both cases." Specifically, we are told that to "Solve for n, given that |n+9|-3n = 3" we can simply set n + 9 = 3 + 3n which gives us n = 3 AND THEN we can set n + 9 = -(3 + 3n) which gives us n = -3.

So following this example, I applied this approach to this problem. Knowing that √x^2 = |x| I went ahead with √((y - 4)^2) as if it were the same thing as |y -4|.
I then went ahead to solve and set y-4 = 4 - y (getting answer y = 4) and then set y - 4 = (-4 - y) (getting no answer) Thus, I moved on to the answer choices with this new equation in mind. Looking for "Is y = 4."
Final answer E.

Can you explain why this approach is the incorrect one to follow from problems such as this one? I understand the logic behind your answer but there is something I am missing because my natural approach would be to follow what I was taught in Chapter 2.

[RonPurewal]

So following this example, I applied this approach to this problem. Knowing that √x^2 = |x| I went ahead with √((y - 4)^2) as if it were the same thing as |y -4|.
I then went ahead to solve and set y-4 = 4 - y (getting answer y = 4) and then set y - 4 = (-4 - y) (getting no answer) Thus, I moved on to the answer choices with this new equation in mind. Looking for "Is y = 4."
Final answer E.

Can you explain why this approach is the incorrect one to follow from problems such as this one? I understand the logic behind your answer but there is something I am missing because my natural approach would be to follow what I was taught in Chapter 2.

Can you explain the genesis of the blue part, please?

You are correct that √((y - 4)^2) is exactly equivalent to |y - 4|.

Your two cases should thus be...
... just y - 4 itself,
... the opposite, which is -(y - 4) = -y + 4 = 4 - y.
You've got the 4 - y.
Instead of just y - 4, though, you have this mysterious (-4 - y) thing, whose origin I can't seem to ascertain.

In any case, regardless of its provenance, that's the part of your work that is in error.
That portion of the solution just reduces to y - 4 = y - 4, and so is true for all numbers in the relevant range.

[RonPurewal]

Ron, thank you for the explanation, this is beginning to make sense to me, you have an amazing ability to explain tough problems in a digestible way.

Thanks. Your kind words are appreciated.

[mafcostanza]

So following this example, I applied this approach to this problem. Knowing that √x^2 = |x| I went ahead with √((y - 4)^2) as if it were the same thing as |y -4|.
I then went ahead to solve and set y-4 = 4 - y (getting answer y = 4) and then set y - 4 = (-4 - y) (getting no answer) Thus, I moved on to the answer choices with this new equation in mind. Looking for "Is y = 4."
Final answer E.

Can you explain why this approach is the incorrect one to follow from problems such as this one? I understand the logic behind your answer but there is something I am missing because my natural approach would be to follow what I was taught in Chapter 2.

Can you explain the genesis of the blue part, please?

Typo. Yes, the blue part should have read " -(4-y) " What I was doing was setting the right hand side of the equation to a negative like the problem in Ch. 2 does. I actually got this far.

Once I got to y - 4 = - (4 - y) [or y - 4 = y - 4] I added 4 to both sides and got y = y. I took this information as useless because it did not reveal anything about y.
So at this point I was left with y=4 [from setting y-4 = 4-y] and y=y [from setting y-4 = -(4-y).
I then went to the answers to see if one told me whether or not y = 4.

Now, logically I can understand how you got to your solution. But using the method in chapter 2 seems to not work here and that is what worries me.

[mafcostanza]

So following this example, I applied this approach to this problem. Knowing that √x^2 = |x| I went ahead with √((y - 4)^2) as if it were the same thing as |y -4|.
I then went ahead to solve and set y-4 = 4 - y (getting answer y = 4) and then set y - 4 = (-4 - y) (getting no answer) Thus, I moved on to the answer choices with this new equation in mind. Looking for "Is y = 4."
Final answer E.

Can you explain why this approach is the incorrect one to follow from problems such as this one? I understand the logic behind your answer but there is something I am missing because my natural approach would be to follow what I was taught in Chapter 2.

Can you explain the genesis of the blue part, please?

Typo. Yes, the blue part should have read " -(4-y) " What I was doing was setting the right hand side of the equation to a negative like the problem in Ch. 2 does. I actually got this far.

Once I got to y - 4 = - (4 - y) [or y - 4 = y - 4] I added 4 to both sides and got y = y. I took this information as useless because it did not reveal anything about y.
So at this point I was left with y=4 [from setting y-4 = 4-y] and y=y [from setting y-4 = -(4-y).
I then went to the answers to see if one told me whether or not y = 4.

Now, logically I can understand how you got to your solution. But using the method in chapter 2 seems to not work here and that is what worries me.

Never mind, I get it now. The "check your answers" part is the one I was skipping. If I just stopped to check the values that would work I can see that the method in Chapter 2 holds up.

Thank you.

[RonPurewal]

Ok. If you have any further questions, fire away.