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mavogler
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M08-16 28 Apr 2020, 15:45
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Loved the question. It really put my mind to work
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M08-16 07 Jun 2020, 09:20
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I think this is a high-quality question and I agree with explanation.
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M08-16 23 Sep 2020, 05:23
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I had trouble understanding the solution at first but then by looking at different cases I think I understood, please let me know if below is right:
we have the simplified equation as |x−3|+√(2−x)+x−3
as "each expression under the square root is greater than or equal to 0" then (2−x) >= 0 which means x=<2
then looking at x−3 this means it will alway be a negative whereas |x−3| will be the same value but positive, therefore they will cancel each other and the equation will be equal to √(2−x)
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M08-16 23 Sep 2020, 08:04
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esitsc
I had trouble understanding the solution at first but then by looking at different cases I think I understood, please let me know if below is right:
we have the simplified equation as |x−3|+√(2−x)+x−3
as "each expression under the square root is greater than or equal to 0" then (2−x) >= 0 which means x=<2
then looking at x−3 this means it will alway be a negative whereas |x−3| will be the same value but positive, therefore they will cancel each other and the equation will be equal to √(2−x)
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Correct. One clarification: \(x \le 2\) means x-3 is negative, so \(|x-3|\) becomes \(|x-3|=-(x-3)=-x+3\).
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M08-16 04 Dec 2021, 04:20
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I think this is a high-quality question and I agree with explanation.
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M08-16 27 Feb 2022, 21:15
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I think this is a high-quality question and I agree with explanation.
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When we plug, say x=1, then sqrt(x^2-6x+9) = sqrt(4) = +2 or -2. Note that the question just says that the expression UNDER the square root should be greater than or equal to 0 which is 4 in this case and it is fine. But, the square root of 4 can be both 2 or -2 & there is no constraint in the question that says, -2 cannot be a solution of the expression. If we consider +2 as the solution, then option A holds i.e. sqrt(2-x) since, x-3=-2 & hence, +2 from the first expression & -2 from this last expression i.e. x-3 cancels out. However, if we consider -2 as the solution of the first expression, then it becomes -2 + sqrt(2-x) -2 = -4 - sqrt(2-x) which makes option B as the correct choice. Therefore, the question definitely has a flaw as per current wording as both A & B can be the possible answer choices.
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M08-16 05 Apr 2022, 01:45
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Devneet
When we plug, say x=1, then sqrt(x^2-6x+9) = sqrt(4) = +2 or -2. Note that the question just says that the expression UNDER the square root should be greater than or equal to 0 which is 4 in this case and it is fine. But, the square root of 4 can be both 2 or -2 & there is no constraint in the question that says, -2 cannot be a solution of the expression. If we consider +2 as the solution, then option A holds i.e. sqrt(2-x) since, x-3=-2 & hence, +2 from the first expression & -2 from this last expression i.e. x-3 cancels out. However, if we consider -2 as the solution of the first expression, then it becomes -2 + sqrt(2-x) -2 = -4 - sqrt(2-x) which makes option B as the correct choice. Therefore, the question definitely has a flaw as per current wording as both A & B can be the possible answer choices.
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This is wrong.

\(\sqrt{...}\) is the square root sign, a function (called the principal square root function), which cannot give negative result. So, this sign (\(\sqrt{...}\)) always means non-negative square root.


The graph of the function f(x) = √x

Notice that it's defined for non-negative numbers and is producing non-negative results.

TO SUMMARIZE:
When the GMAT (and generally in math) provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the non-negative root. That is:

\(\sqrt{9} = 3\), NOT +3 or -3;
\(\sqrt[4]{16} = 2\), NOT +2 or -2;

Notice that in contrast, the equation \(x^2 = 9\) has TWO solutions, +3 and -3. Because \(x^2 = 9\) means that \(x =-\sqrt{9}=-3\) or \(x=\sqrt{9}=3\).
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M08-16 17 Apr 2022, 04:50
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I think this is a high-quality question and I agree with explanation.
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M08-16 08 May 2022, 01:39
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I think this is a high-quality question and I agree with explanation. Great question with a very important insight!
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I read somewhere that in questions where the equations are equated, plugging a certain number (after giving due consideration to the constraints) will ALWAYS give the same result from both the matching equations.

In this question, I simply plugged in "0" and A gave the same result.

Bunuel Please correct me if my understanding is incorrect for such equating questions. Did I get lucky in this question? Can I always plug in values while solving "equating equations" questions?

I have another question related to the explanation provided for this question.

The following Absolute Property is provided in the explanation

When x≤0 then |x|=−x, or more generally when some expression ≤0 then |some expression|=−(some expression). For example: |−5|=5=−(−5);

But, in the following step, we get x≤2 and not x≤0. Are we supposed to only consider less than or equal to sign to consider -(-x) ??

Now, as the expressions under the square roots are more than or equal to zero then 2−x≥0 → x≤2. Next: as x≤2 then |x−3| becomes |x−3|=−(x−3)=−x+3


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Sud
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M08-16 14 Nov 2022, 22:21
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Sud2021
I read somewhere that in questions where the equations are equated, plugging a certain number (after giving due consideration to the constraints) will ALWAYS give the same result from both the matching equations.

In this question, I simply plugged in "0" and A gave the same result.

Bunuel Please correct me if my understanding is incorrect for such equating questions. Did I get lucky in this question? Can I always plug in values while solving "equating equations" questions?

I have another question related to the explanation provided for this question.

The following Absolute Property is provided in the explanation

When x≤0 then |x|=−x, or more generally when some expression ≤0 then |some expression|=−(some expression). For example: |−5|=5=−(−5);

But, in the following step, we get x≤2 and not x≤0. Are we supposed to only consider less than or equal to sign to consider -(-x) ??

Now, as the expressions under the square roots are more than or equal to zero then 2−x≥0 → x≤2. Next: as x≤2 then |x−3| becomes |x−3|=−(x−3)=−x+3


Regards,
Sud
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1. Yes, you can plug x = 0, to het the answer.

2. We get \(x \leq 2\), from \(\sqrt{2 - x} \):

The expressions under the square roots are more than or equal to zero then \(2-x \ge 0\) \(\rightarrow\) \(x \le 2\).
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M08-16 04 Mar 2023, 11:36
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I think this is a high-quality question and I agree with explanation.
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M08-16 01 Jul 2023, 04:38
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If the roots are equal to or greater than 0, then shouldn't |x-3| also be positive or equal to zero.
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M08-16 01 Jul 2023, 04:50
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Vaishali2004
If the roots are equal to or greater than 0, then shouldn't |x-3| also be positive or equal to zero.
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The absolute value is, in a sense, the measure of a distance, so it is always greater than or equal to 0. For instance, |x - 3| represents the distance between x and 3 on the number line.

10. Absolute Value



For more check Ultimate GMAT Quantitative Megathread



Hope it helps.
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M08-16 01 Jul 2023, 05:04
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Right, but we can just as well say that x-3 is greater than 0,therefore x is greater than 3, right?

Posted from my mobile device
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M08-16 01 Jul 2023, 08:25
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Vaishali2004
Right, but we can just as well say that x-3 is greater than 0,therefore x is greater than 3, right?

Posted from my mobile device
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Honestly, I'm finding it difficult to grasp your point or determine your argument's direction... Generally, x - 3 can be negative, zero, or positive. But in the context of this question, x - 3 is negative, since we've established x <= 2. Therefore, |x - 3| = -(x - 3) = 3 - x.
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M08-16 09 Aug 2023, 10:32
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I think this is a high-quality question and I agree with explanation.
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M08-16 12 Aug 2023, 10:56
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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M08-16 16 Jun 2024, 05:48
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I think this is a high-quality question.
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