M08-16

Correct. One clarification: \(x \le 2\) means x-3 is negative, so \(|x-3|\) becomes \(|x-3|=-(x-3)=-x+3\).

I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation.

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.

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 \(x=\sqrt{9}=3\).

I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation. Great question with a very important insight!

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

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 \(2-x \ge 0\) \(\rightarrow\) \(x \le 2\).

I think this is a high-quality question and I agree with explanation.

If the roots are equal to or greater than 0, then shouldn't |x-3| also be positive or equal to zero.

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

• Theory
  Math: Absolute value (Modulus)
  Absolute Value: Tips and hints
  Properties of Absolute Values on the GMAT
  Absolute modulus : A better understanding
  GMAT Question Patterns: Abolute Value
  The Reason Behind Absolute Value Questions on the GMAT
  
  • Questions
    Inequality and absolute value questions from my collection
    The E-GMAT Question Series on ABSOLUTE VALUE
    DS Questions
    PS Questions

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.

Right, but we can just as well say that x-3 is greater than 0,therefore x is greater than 3, right?

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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.

I think this is a high-quality question and I agree with explanation.

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

I think this is a high-quality question.