Let the function g be defined as \(\mathrm{g(x) = \frac{|x - a| + 45}{3a}}\), where a is a positive constant....

1. INFER the solution strategy

• Given: \(\mathrm{g(x) = \frac{|x - a| + 45}{3a}}\) and \(\mathrm{g(a) = 6}\)
• To find \(\mathrm{g(10)}\), we first need to determine the value of constant \(\mathrm{a}\)
• Strategy: Use the condition \(\mathrm{g(a) = 6}\) to solve for \(\mathrm{a}\), then evaluate \(\mathrm{g(10)}\)

2. INFER the key absolute value property

• When \(\mathrm{x = a}\) in the expression \(\mathrm{|x - a|}\), we get \(\mathrm{|a - a| = 0}\)
• This simplifies \(\mathrm{g(a)}\) significantly

3. SIMPLIFY to find the value of a

• Substitute \(\mathrm{x = a}\) into the function:

\(\mathrm{g(a) = \frac{|a - a| + 45}{3a}}\)
\(\mathrm{= \frac{0 + 45}{3a}}\)
\(\mathrm{= \frac{45}{3a}}\)
\(\mathrm{= \frac{15}{a}}\)

• Since \(\mathrm{g(a) = 6}\): \(\mathrm{\frac{15}{a} = 6}\)
• Solve: \(\mathrm{15 = 6a}\), so \(\mathrm{a = \frac{15}{6} = 2.5}\)

4. SIMPLIFY to evaluate g(10)

• Now substitute \(\mathrm{x = 10}\) and \(\mathrm{a = 2.5}\):

\(\mathrm{g(10) = \frac{|10 - 2.5| + 45}{3 \times 2.5}}\)

• Calculate: \(\mathrm{g(10) = \frac{7.5 + 45}{7.5} = \frac{52.5}{7.5} = 7.0}\)

Answer: D) 7.0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students try to evaluate \(\mathrm{g(10)}\) directly without first finding the value of \(\mathrm{a}\).

They might attempt to work with \(\mathrm{g(10) = \frac{|10 - a| + 45}{3a}}\) as is, not realizing they need to determine \(\mathrm{a}\) first using the given condition. This leads to an expression with two unknowns (the result and \(\mathrm{a}\)) which cannot be solved. This causes them to get stuck and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need to find \(\mathrm{a}\) first, but make arithmetic errors in the calculation.

For example, they might incorrectly compute \(\mathrm{52.5/7.5}\), getting \(\mathrm{7.5}\) instead of \(\mathrm{7.0}\), or make errors when solving \(\mathrm{15/a = 6}\). This may lead them to select Choice C (6.5) due to calculation mistakes.

The Bottom Line:

This problem requires recognizing the two-stage solution process: first determine the parameter, then use it to evaluate the target expression. The absolute value property \(\mathrm{|a - a| = 0}\) is the key insight that makes the first stage solvable.