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What is the greatest integer that is less than: \(-|\sqrt{3}-3|\) ?
Let's break down what we're asked to find. We have the expression \(-|\sqrt{3} - 3|\) and we need to find the greatest integer that is less than this value.
Think of it this way: if someone gives you a number like -1.7, the greatest integer less than -1.7 would be -2 (since -2 is smaller than -1.7, but -1 is actually bigger than -1.7).
So our job is to: first calculate what \(-|\sqrt{3} - 3|\) equals, then find the largest whole number that's smaller than that result.
Process Skill: TRANSLATE
Let's think about \(\sqrt{3}\) compared to 3. We know that:
Since 3 is between 1 and 4, we know that \(\sqrt{3}\) is between 1 and 2. More precisely, \(\sqrt{3} \approx 1.732\).
This means \(\sqrt{3}\) is definitely less than 3. In fact, \(\sqrt{3} - 3 = 1.732 - 3 = -1.268\) (approximately).
So the expression inside our absolute value bars, \((\sqrt{3} - 3)\), is negative.
Now we need to handle \(|\sqrt{3} - 3|\). Since we determined that \(\sqrt{3} - 3\) is negative, the absolute value will make it positive.
Remember: the absolute value of a negative number removes the negative sign.
So \(|\sqrt{3} - 3| = |\text{negative number}| = -(\sqrt{3} - 3) = -(\sqrt{3}) + 3 = 3 - \sqrt{3}\)
Therefore: \(|\sqrt{3} - 3| = 3 - \sqrt{3}\)
Now we can evaluate our original expression:
\(-|\sqrt{3} - 3| = -(3 - \sqrt{3}) = -3 + \sqrt{3} = \sqrt{3} - 3\)
Using \(\sqrt{3} \approx 1.732\):
\(\sqrt{3} - 3 \approx 1.732 - 3 = -1.268\)
So our expression \(-|\sqrt{3} - 3|\) equals approximately -1.268.
We need the greatest integer less than -1.268.
Let's think about the integers around -1.268:
Therefore, the greatest integer less than -1.268 is -2.
The greatest integer that is less than \(-|\sqrt{3} - 3|\) is -2.
This matches answer choice B.
Many students confuse "greatest integer less than" with "greatest integer less than or equal to" (floor function). For a negative number like -1.268, they might think the answer is -1 instead of -2, forgetting that -1 is actually greater than -1.268 on the number line.
Students may struggle with the nested structure of \(-|\sqrt{3} - 3|\) and lose track of the negative sign outside the absolute value bars. They might attempt to evaluate \(|\sqrt{3} - 3|\) first without properly accounting for the negative sign that precedes it.
When students determine that \(\sqrt{3} - 3\) is negative, they may incorrectly apply the absolute value. Instead of getting \(|\sqrt{3} - 3| = 3 - \sqrt{3}\), they might mistakenly write \(|\sqrt{3} - 3| = \sqrt{3} - 3\), forgetting that absolute value makes negative expressions positive.
Students might use an incorrect approximation for \(\sqrt{3}\) (such as 1.5 instead of 1.732) or make arithmetic errors when calculating \(3 - \sqrt{3}\) or \(\sqrt{3} - 3\), leading to an incorrect final numerical value.
After correctly finding \(|\sqrt{3} - 3| = 3 - \sqrt{3}\), students may make sign errors when applying the negative sign outside, potentially getting \(3 - \sqrt{3}\) instead of the correct \(\sqrt{3} - 3 = -1.268\).
Even after correctly calculating that the expression equals approximately -1.268, students may select -1 as their answer because they incorrectly think -1 is less than -1.268, not recognizing that on the number line, -2 is to the left of -1.268, making -2 the correct "greatest integer less than" the target value.