If b and (2x - 3b = 0, which of the following must be true?

1. Translate the problem requirements: We have two pieces of information: \(\mathrm{b}\) is less than 2, and there's an equation \(\mathrm{2x - 3b = 0}\) that relates \(\mathrm{x}\) and \(\mathrm{b}\). We need to find what must be true about \(\mathrm{x}\) given these constraints.
2. Express x in terms of b: Solve the equation \(\mathrm{2x - 3b = 0}\) to find how \(\mathrm{x}\) depends on \(\mathrm{b}\).
3. Apply the constraint on b: Use the given condition \(\mathrm{b < 2}\) to determine what this means for \(\mathrm{x}\) through the relationship we found.
4. Evaluate answer choices: Check which statement about \(\mathrm{x}\) must always be true given our constraint.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're given in everyday terms. We have a number called b that we know is smaller than 2. We also have an equation that connects two variables: \(\mathrm{2x - 3b = 0}\). Think of this equation as a rule that tells us how x and b are related to each other.

Our job is to figure out what we can say for certain about the value of x, given that we know b is less than 2.

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

2. Express x in terms of b

Now let's solve the equation \(\mathrm{2x - 3b = 0}\) to see exactly how x depends on b.

Starting with: \(\mathrm{2x - 3b = 0}\)
We want to get x by itself, so let's add \(\mathrm{3b}\) to both sides:
\(\mathrm{2x = 3b}\)

Now divide both sides by 2:
\(\mathrm{x = \frac{3b}{2}}\)

This tells us that x is always equal to three-halves times whatever b is. So if we know something about b, we can figure out something about x.

3. Apply the constraint on b

We know that \(\mathrm{b < 2}\). Since \(\mathrm{x = \frac{3b}{2}}\), we can substitute this constraint.

If \(\mathrm{b < 2}\), then when we multiply both sides by \(\mathrm{\frac{3}{2}}\) (which is positive), the inequality direction stays the same:

\(\mathrm{\frac{3b}{2} < \frac{3(2)}{2}}\)
\(\mathrm{\frac{3b}{2} < \frac{6}{2}}\)
\(\mathrm{\frac{3b}{2} < 3}\)

But remember, \(\mathrm{x = \frac{3b}{2}}\), so this means:
\(\mathrm{x < 3}\)

Process Skill: APPLY CONSTRAINTS - Using the given limitation on b to determine what must be true about x

4. Evaluate answer choices

Now let's check our result \(\mathrm{x < 3}\) against the answer choices:

1. \(\mathrm{x > -3}\): This might be true, but it's not necessarily true for all possible values
2. \(\mathrm{x < 2}\): This is too restrictive - x could be between 2 and 3
3. \(\mathrm{x = 3}\): This would only be true if \(\mathrm{b = 2}\), but we know \(\mathrm{b < 2}\)
4. \(\mathrm{x < 3}\): This matches exactly what we found!
5. \(\mathrm{x > 3}\): This contradicts our finding

Final Answer

The answer is (D) \(\mathrm{x < 3}\).

This must be true because we established that \(\mathrm{x = \frac{3b}{2}}\), and since \(\mathrm{b < 2}\), we get \(\mathrm{x < \frac{3(2)}{2} = 3}\). The inequality \(\mathrm{x < 3}\) captures exactly what we can conclude about x given the constraint on b.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Missing the constraint on b
Students often focus solely on the equation \(\mathrm{2x - 3b = 0}\) and forget that we're also given the crucial constraint \(\mathrm{b < 2}\). They might solve for x in terms of b correctly but then fail to use the constraint to determine what must be true about x. This leads them to think they don't have enough information to answer the question.

Faltering Point 2: Misunderstanding what "must be true" means
Students might confuse this with finding a specific value of x rather than understanding they need to find a relationship or constraint that will always hold true given the conditions. This can lead them to look for an exact value rather than an inequality.

Errors while executing the approach

Faltering Point 1: Inequality direction errors when multiplying
When going from \(\mathrm{b < 2}\) to \(\mathrm{x < 3}\), students need to multiply both sides by \(\mathrm{\frac{3}{2}}\). Since \(\mathrm{\frac{3}{2}}\) is positive, the inequality direction stays the same. However, students sometimes get confused about when to flip inequality signs and might incorrectly reverse the direction, leading to \(\mathrm{x > 3}\) instead of \(\mathrm{x < 3}\).

Faltering Point 2: Arithmetic errors in the constraint application
Students might make calculation mistakes when computing \(\mathrm{\frac{3(2)}{2} = 3}\). Simple arithmetic errors like getting \(\mathrm{\frac{6}{2} = 4}\) or other computational mistakes can lead to wrong final inequalities and incorrect answer choices.

Errors while selecting the answer

Faltering Point 1: Choosing a less restrictive but true option
Students who correctly find \(\mathrm{x < 3}\) might see that option (A) \(\mathrm{x > -3}\) could also be true and select it instead. They fail to recognize that the question asks for what "must be true," and while \(\mathrm{x > -3}\) might often be true, \(\mathrm{x < 3}\) is the tightest and most precise constraint that must always hold given our conditions.

Alternate Solutions

Smart Numbers Approach

Instead of working algebraically, we can test specific values of b that satisfy our constraint and see what happens to x.

Step 1: Set up the relationship
From the equation \(\mathrm{2x - 3b = 0}\), we can solve for x:
\(\mathrm{2x = 3b}\)
\(\mathrm{x = \frac{3b}{2}}\)

Step 2: Choose smart values for b
Since \(\mathrm{b < 2}\), let's test several values that satisfy this constraint:

Test 1: Let \(\mathrm{b = 1}\)
\(\mathrm{x = \frac{3(1)}{2} = 1.5}\)
Since \(\mathrm{1.5 < 3}\), this supports answer choice (D)

Test 2: Let \(\mathrm{b = 0}\)
\(\mathrm{x = \frac{3(0)}{2} = 0}\)
Since \(\mathrm{0 < 3}\), this supports answer choice (D)

Test 3: Let \(\mathrm{b = -1}\)
\(\mathrm{x = \frac{3(-1)}{2} = -1.5}\)
Since \(\mathrm{-1.5 < 3}\), this supports answer choice (D)

Test 4: Let b approach 2 (but stay less than 2)
Let \(\mathrm{b = 1.99}\)
\(\mathrm{x = \frac{3(1.99)}{2} = 2.985}\)
Since \(\mathrm{2.985 < 3}\), this still supports answer choice (D)

Step 3: Analyze the pattern
In every case where \(\mathrm{b < 2}\), we consistently get \(\mathrm{x < 3}\). Even as b approaches its maximum allowed value (just under 2), x approaches but never reaches 3.

Step 4: Verify against other answer choices

• (A) \(\mathrm{x > -3}\): False when \(\mathrm{b = -5}\) (which gives \(\mathrm{x = -7.5}\))
• (B) \(\mathrm{x < 2}\): False when \(\mathrm{b = 1.5}\) (which gives \(\mathrm{x = 2.25}\))
• (C) \(\mathrm{x = 3}\): Never true since b never equals 2
• (E) \(\mathrm{x > 3}\): Never true since our tests show x is always less than 3

Therefore, the answer is (D) \(\mathrm{x < 3}\).