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If \(\mathrm{a}\) and \(\mathrm{b}\) are, respectively, the least and greatest nonnegative real number solutions to \(\left|\frac{\mathrm{x} - 6}{5}\right| \leq 5.5\), what is the value of \(\mathrm{b} - \mathrm{a}\)?
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:
Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding
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: \(-5.5 \leq \frac{\mathrm{x} - 6}{5} \leq 5.5\)
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.
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 \(\mathrm{x} \in [-21.5, 33.5]\).
The problem specifically asks for nonnegative real number solutions, meaning \(\mathrm{x} \geq 0\).
Looking at our range \([-21.5, 33.5]\):
Process Skill: APPLY CONSTRAINTS - Ensuring we only consider solutions that meet all problem requirements
Therefore:
The value of b - a is 33.5.
This matches answer choice D. 33.5.
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.
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.
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.
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 \(27.5 + 6 = 33.5\).
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.
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.
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 \(\mathrm{b} - \mathrm{a} = 33.5 - 0 = 33.5\).