Finding Maximum Value: Comparing Expressions with x > 0

Q: Why can't we just ignore the variable a and compare the expressions?

Because the sign and value of a completely changes the ranking! If $ a = 1 $, then $ 300ax $ is huge. But if $ a = -1 $, it becomes a large negative number.

Q: What if we assume a is positive?

Assumptions aren't allowed unless stated in the problem! The question doesn't give us any information about $ a $, so we must work with what we know: only that $ x > 0 $.

Q: Can we at least say that -2x is definitely the smallest?

Yes! Since $ x > 0 $, we know $ -2x $ is definitely negative. But we still can't determine which of the other expressions is largest without knowing $ a $.

Q: How do I recognize when a problem has insufficient information?

Look for unknown variables or parameters that aren't defined. If comparing expressions depends on values you don't know, then "insufficient information" is likely the answer.

Q: What's the difference between this and regular expression comparison?

Regular comparison: All variables are known or definedThis type: Contains undefined parameters that affect the outcomeAlways check if you have enough information before attempting to solve!

Expression Comparison with Unknown Parameters

Since $0 < x$

Which of the values is the largest?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the largest value

00:08 The unknown is necessarily positive, and the number is also positive

00:18 However, we don't know if A is positive or negative

00:28 Therefore we cannot know

00:34 The same thing happens here too

00:54 Negative times positive is always negative

01:00 This option is necessarily not the largest

01:05 Therefore we cannot know, and this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Since $0 < x$

Which of the values is the largest?

Step-by-step solution

To solve this problem, we'll evaluate the expressions based on the condition $0 < x$ :

If $300ax$ is considered, the sign and magnitude depend heavily on $a$ . If $a$ is positive, $300ax$ is large positive. If $a$ is negative, it's large negative.
$\frac{x}{300a}$ is affected by both $x$ and $a$ . If $a$ is positive, this expression shrinks, if $a$ is negative, the sign is inverted, and it becomes positive, but its magnitude determination needs $a$ .
$-2x$ results in a definite negative value because $x$ is positive.

Without knowing the sign or value of $a$ , we can't definitively compare or compute the sizes of the expressions relative to each other.

Therefore, the solution to the problem is: It is not possible to calculate.

Final Answer

It is not possible to calculate

Key Points to Remember

Essential concepts to master this topic

Analysis Rule: Examine each expression for dependence on unknown variables
Technique: Test cases: if $a > 0$ , then $300ax$ is largest; if $a < 0$ , it's smallest
Check: Verify all expressions contain unknown $a$ making definitive comparison impossible ✓

Common Mistakes

Avoid these frequent errors

Assuming expressions can be compared without knowing all variables
Don't try to rank $300ax$ and $\frac{x}{300a}$ without knowing the sign of $a$ = wrong conclusion! The value and sign of $a$ completely changes which expression is largest. Always identify all unknown parameters before attempting comparisons.

Practice Quiz

Test your knowledge with interactive questions

What will be the sign of the result of the next exercise?

$(-2)\cdot(-4)=$

FAQ

Everything you need to know about this question

Why can't we just ignore the variable a and compare the expressions?

Because the sign and value of a completely changes the ranking! If $a = 1$ , then $300ax$ is huge. But if $a = -1$ , it becomes a large negative number.

What if we assume a is positive?

Assumptions aren't allowed unless stated in the problem! The question doesn't give us any information about $a$ , so we must work with what we know: only that $x > 0$ .

Can we at least say that -2x is definitely the smallest?

Yes! Since $x > 0$ , we know $-2x$ is definitely negative. But we still can't determine which of the other expressions is largest without knowing $a$ .

How do I recognize when a problem has insufficient information?

Look for unknown variables or parameters that aren't defined. If comparing expressions depends on values you don't know, then "insufficient information" is likely the answer.

What's the difference between this and regular expression comparison?

Regular comparison: All variables are known or defined
This type: Contains undefined parameters that affect the outcome

Always check if you have enough information before attempting to solve!

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