What is the greatest integer that is less than: -{|sqrt(3)-3|} ?

1. Translate the problem requirements: We need to find the greatest integer less than the expression \(-|\sqrt{3} - 3|\). This means we need to evaluate the expression first, then find the largest whole number that is smaller than our result.
2. Determine the relationship between \(\sqrt{3}\) and 3: Compare these values to understand what's inside the absolute value bars and whether the result will be positive or negative.
3. Simplify the absolute value expression: Use the relationship from step 2 to remove the absolute value bars and get a concrete numerical expression.
4. Approximate the final expression: Calculate or estimate the value to determine which integer range it falls into.
5. Identify the greatest integer less than our result: Find the largest whole number that is smaller than the value from step 4.

Execution of Strategic Approach

1. Translate the problem requirements

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

2. Determine the relationship between \(\sqrt{3}\) and 3

Let's think about \(\sqrt{3}\) compared to 3. We know that:

• \(\sqrt{1} = 1\)
• \(\sqrt{4} = 2\)
• \(\sqrt{9} = 3\)

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.

3. Simplify the absolute value expression

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: \(|\sqrt{3} - 3| = 3 - \sqrt{3}\)

4. Approximate the final expression

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.

5. Identify the greatest integer less than our result

We need the greatest integer less than -1.268.

Let's think about the integers around -1.268:

• -1 is greater than -1.268 (since -1 = -1.000)
• -2 is less than -1.268 (since -2 = -2.000)
• -3 is also less than -1.268, but it's not the greatest

Therefore, the greatest integer less than -1.268 is -2.

Final Answer

The greatest integer that is less than \(-|\sqrt{3} - 3|\) is -2.

This matches answer choice B.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "greatest integer less than" means

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.

2. Incorrect setup of the absolute value expression

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.

Errors while executing the approach

1. Incorrectly handling the absolute value of a negative expression

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.

2. Approximation errors with \(\sqrt{3}\)

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.

3. Sign errors when simplifying \(-|\sqrt{3} - 3|\)

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 \(\sqrt{3} - 3 = -1.268\).

Errors while selecting the answer

1. Choosing the wrong integer due to number line confusion

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.