If a and b are, respectively, the least and greatest nonnegative real number solutions to left|(x - 6)/5right| leq 5.5,...

1. Translate the problem requirements: We need to find all nonnegative real numbers x that satisfy the absolute value inequality, then identify the smallest value (a) and largest value (b), and calculate their difference.
2. Interpret the absolute value inequality as a distance constraint: The expression \(|\frac{\mathrm{x}-6}{5}| \leq 5.5\) represents all values where the distance from \(\frac{\mathrm{x}-6}{5}\) to zero is at most 5.5 units.
3. Convert the inequality to a compound inequality: Remove the absolute value by creating the equivalent double inequality that captures all values within the distance constraint.
4. Solve for the complete range of x values: Algebraically manipulate the compound inequality to find the interval of all possible x values.
5. Apply the nonnegative constraint and calculate the difference: Identify which endpoints of our interval satisfy \(\mathrm{x} \geq 0\), then find b - a.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to find. We have an inequality involving absolute value: \(|\frac{\mathrm{x} - 6}{5}| \leq 5.5\), and we need to find all nonnegative real number solutions.

In everyday language, we're looking for:

• All values of x that make this inequality true AND are greater than or equal to zero
• The smallest such value (we'll call this 'a')
• The largest such value (we'll call this 'b')
• Then we calculate b - a

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

2. Interpret the absolute value inequality as a distance constraint

Think of absolute value as measuring distance. When we see \(|\frac{\mathrm{x} - 6}{5}| \leq 5.5\), we can read this as:
"The distance from \(\frac{\mathrm{x} - 6}{5}\) to zero is at most 5.5 units."

Imagine a number line where we place the point \(\frac{\mathrm{x} - 6}{5}\). This point can be anywhere from 5.5 units to the left of zero up to 5.5 units to the right of zero. So \(\frac{\mathrm{x} - 6}{5}\) can range from -5.5 to +5.5.

Mathematically, this gives us: \(-5.5 \leq \frac{\mathrm{x} - 6}{5} \leq 5.5\)

3. Convert the inequality to a compound inequality

We've established that our absolute value inequality \(|\frac{\mathrm{x} - 6}{5}| \leq 5.5\) is equivalent to the compound inequality:
\(-5.5 \leq \frac{\mathrm{x} - 6}{5} \leq 5.5\)

This compound inequality captures all values where \(\frac{\mathrm{x} - 6}{5}\) is within 5.5 units of zero in either direction.

4. Solve for the complete range of x values

Now we solve the compound inequality \(-5.5 \leq \frac{\mathrm{x} - 6}{5} \leq 5.5\) for x.

We can work with both parts simultaneously by multiplying everything by 5:
\(-5.5 \times 5 \leq (\mathrm{x} - 6) \leq 5.5 \times 5\)
\(-27.5 \leq \mathrm{x} - 6 \leq 27.5\)

Next, we add 6 to all parts:
\(-27.5 + 6 \leq \mathrm{x} \leq 27.5 + 6\)
\(-21.5 \leq \mathrm{x} \leq 33.5\)

So our complete range of solutions is \(\mathrm{x} \in [-21.5, 33.5]\).

5. Apply the nonnegative constraint and calculate the difference

The problem specifically asks for nonnegative real number solutions, meaning \(\mathrm{x} \geq 0\).

Looking at our range \([-21.5, 33.5]\):

• The left endpoint -21.5 is negative, so it doesn't qualify
• We need \(\mathrm{x} \geq 0\), so our actual range of valid solutions is \([0, 33.5]\)

Process Skill: APPLY CONSTRAINTS - Ensuring we only consider solutions that meet all problem requirements

Therefore:

• a = 0 (the least nonnegative solution)
• b = 33.5 (the greatest nonnegative solution)
• \(\mathrm{b} - \mathrm{a} = 33.5 - 0 = 33.5\)

Final Answer

The value of b - a is 33.5.

This matches answer choice D. 33.5.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the absolute value inequality structure

Students often struggle with recognizing that \(|\frac{\mathrm{x} - 6}{5}| \leq 5.5\) needs to be converted to a compound inequality. They might try to solve it by simply removing the absolute value bars without considering both positive and negative cases, leading to an incomplete solution.

2. Overlooking the nonnegative constraint from the start

Many students focus only on solving the absolute value inequality and forget that the problem specifically asks for "nonnegative real number solutions" \((\mathrm{x} \geq 0)\). This constraint is crucial for determining the correct values of a and b, but students often remember it only at the end or forget it entirely.

3. Misinterpreting what a and b represent

Students may confuse which values they need to find. The problem asks for the "least and greatest nonnegative real number solutions," but some students might think they need the least and greatest values from the entire solution set (including negative values) rather than specifically from the nonnegative subset.

Errors while executing the approach

1. Sign errors when solving the compound inequality

When converting \(|\frac{\mathrm{x} - 6}{5}| \leq 5.5\) to \(-5.5 \leq \frac{\mathrm{x} - 6}{5} \leq 5.5\), students frequently make sign errors, especially when multiplying by 5 or adding 6. Common mistakes include getting \(-27.5 + 6 = -21.5\) wrong or miscalculating \(27.5 + 6 = 33.5\).

2. Incorrectly applying the nonnegative constraint

Even when students remember the \(\mathrm{x} \geq 0\) constraint, they might apply it incorrectly. For example, they might think the constraint eliminates the entire left portion of their solution interval, or they might incorrectly modify their equation rather than simply restricting their final answer range.

Errors while selecting the answer

1. Calculating b - a using wrong values

Students might correctly find that the complete solution range is \([-21.5, 33.5]\) but then use a = -21.5 instead of a = 0 when calculating b - a. This would give them \(\mathrm{b} - \mathrm{a} = 33.5 - (-21.5) = 55\), leading them to incorrectly choose answer choice E instead of the correct answer D.

2. Confusing the range length with the actual difference requested

Some students might calculate the total length of the solution interval \([-21.5, 33.5]\), which would be \(33.5 - (-21.5) = 55\), instead of focusing specifically on the nonnegative portion and calculating \(\mathrm{b} - \mathrm{a} = 33.5 - 0 = 33.5\).