If \(0 , which of the following inequalities must be true ? \(\mathrm{x}^5 \(\mathrm{x}^4 + \mathrm{x}^5 \(\mathrm{x}^4 - \mathrm{x}^5...

1. Translate the problem requirements: We need to determine which inequalities are always true when x is a fraction between 0 and 1 (exclusive). The inequalities compare different powers of x.
2. Establish the key property of fractions under repeated multiplication: Understand how multiplying a fraction by itself makes it smaller, and use this to compare different powers of x.
3. Apply the power comparison principle to each inequality: For each statement, determine which side has the larger total power and use this to establish the direction of the inequality.
4. Verify with concrete examples: Test with a specific value like x = 0.5 to confirm our reasoning and ensure all three inequalities hold.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked. We have a number x that's between 0 and 1 - think of it as a fraction like \(\frac{1}{2}\), \(\frac{1}{3}\), or 0.7. We need to figure out which of three inequalities will ALWAYS be true for ANY such fraction.

The three statements compare different powers of this fraction:

• I. \(\mathrm{x}^5 < \mathrm{x}^3\) (Is x to the 5th power smaller than x to the 3rd power?)
• II. \(\mathrm{x}^4 + \mathrm{x}^5 < \mathrm{x}^3 + \mathrm{x}^2\) (Is the sum of the 4th and 5th powers smaller than the sum of the 3rd and 2nd powers?)
• III. \(\mathrm{x}^4 - \mathrm{x}^5 < \mathrm{x}^2 - \mathrm{x}^3\) (Is this difference of higher powers smaller than this difference of lower powers?)

Process Skill: TRANSLATE - Converting the abstract inequality symbols into plain English helps us understand what we're comparing

2. Establish the key property of fractions under repeated multiplication

Here's the key insight: When you multiply a fraction by itself, it gets smaller. Let's see this with a concrete example using \(\mathrm{x} = \frac{1}{2}\):

Starting with \(\frac{1}{2}\):

• \(\mathrm{x} = \frac{1}{2} = 0.5\)
• \(\mathrm{x}^2 = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) = \frac{1}{4} = 0.25\)
• \(\mathrm{x}^3 = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) = \frac{1}{8} = 0.125\)
• \(\mathrm{x}^4 = \frac{1}{16} = 0.0625\)
• \(\mathrm{x}^5 = \frac{1}{32} = 0.03125\)

Notice the pattern: 0.5 > 0.25 > 0.125 > 0.0625 > 0.03125

In plain English: The more times you multiply a fraction by itself, the smaller it becomes. So higher powers of fractions are always smaller than lower powers.

This gives us the general rule: If \(0 < \mathrm{x} < 1\), then \(\mathrm{x}^a > \mathrm{x}^b\) whenever \(\mathrm{a} < \mathrm{b}\)

3. Apply the power comparison principle to each inequality

Now let's check each statement using our understanding:

Statement I: \(\mathrm{x}^5 < \mathrm{x}^3\)

Since 5 > 3, and higher powers of fractions are smaller, \(\mathrm{x}^5\) must indeed be smaller than \(\mathrm{x}^3\). ✓ TRUE

Statement II: \(\mathrm{x}^4 + \mathrm{x}^5 < \mathrm{x}^3 + \mathrm{x}^2\)

Let's compare term by term:

• \(\mathrm{x}^4 < \mathrm{x}^2\) (since 4 > 2)
• \(\mathrm{x}^5 < \mathrm{x}^3\) (since 5 > 3)

When we add smaller numbers to smaller numbers, we get a smaller sum. So \(\mathrm{x}^4 + \mathrm{x}^5 < \mathrm{x}^2 + \mathrm{x}^3\). ✓ TRUE

Statement III: \(\mathrm{x}^4 - \mathrm{x}^5 < \mathrm{x}^2 - \mathrm{x}^3\)

This one is trickier. Let's rearrange it by moving terms around:

\(\mathrm{x}^4 - \mathrm{x}^5 < \mathrm{x}^2 - \mathrm{x}^3\)

Add \(\mathrm{x}^5\) and \(\mathrm{x}^3\) to both sides:

\(\mathrm{x}^4 + \mathrm{x}^3 < \mathrm{x}^2 + \mathrm{x}^5\)

Rearrange:

\(\mathrm{x}^4 - \mathrm{x}^2 < \mathrm{x}^5 - \mathrm{x}^3\)

Factor:

\(\mathrm{x}^2(\mathrm{x}^2 - 1) < \mathrm{x}^3(\mathrm{x}^2 - 1)\)

Since \(0 < \mathrm{x} < 1\), we know that \((\mathrm{x}^2 - 1)\) is negative. When we divide both sides by this negative number, the inequality flips:

\(\mathrm{x}^2 > \mathrm{x}^3\)

This is true since 2 < 3. ✓ TRUE

Process Skill: MANIPULATE - Algebraically rearranging the inequality helped us see the underlying relationship

4. Verify with concrete examples

Let's double-check with x = 0.5:

Statement I: 0.03125 < 0.125 ✓

Statement II: 0.0625 + 0.03125 = 0.09375 and 0.125 + 0.25 = 0.375

Indeed, 0.09375 < 0.375 ✓

Statement III: 0.0625 - 0.03125 = 0.03125 and 0.25 - 0.125 = 0.125

Indeed, 0.03125 < 0.125 ✓

All three statements are verified to be true.

Final Answer

All three inequalities (I, II, and III) must be true when \(0 < \mathrm{x} < 1\).

The answer is E. I, II and III

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the constraint \(0 < \mathrm{x} < 1\)

Students often fail to fully grasp what \(0 < \mathrm{x} < 1\) means in practical terms. They might think of it as "any number between 0 and 1" without understanding that this specifically refers to proper fractions or decimals like 0.3, 0.7, 0.99, etc. This leads them to incorrectly apply rules for integers or numbers greater than 1, where higher powers would be larger, not smaller.

2. Not recognizing the fundamental property of fraction powers

Many students don't immediately realize that when \(0 < \mathrm{x} < 1\), higher powers become progressively smaller (\(\mathrm{x} > \mathrm{x}^2 > \mathrm{x}^3 > \mathrm{x}^4 > \mathrm{x}^5\)). They might incorrectly assume that \(\mathrm{x}^5 > \mathrm{x}^3\) because 5 > 3, applying integer logic instead of fraction logic. This fundamental misunderstanding derails their entire approach to comparing the inequalities.

3. Choosing to test specific values instead of proving generally

Students often jump straight to testing specific values like x = 0.5 without first establishing the general principle. While testing can verify answers, relying solely on examples doesn't prove that the inequalities are true for ALL values where \(0 < \mathrm{x} < 1\). This approach might work for simpler statements but fails for complex ones like Statement III.

Errors while executing the approach

1. Sign errors when manipulating Statement III

Statement III requires algebraic manipulation: \(\mathrm{x}^4 - \mathrm{x}^5 < \mathrm{x}^2 - \mathrm{x}^3\). When rearranging and factoring to get \(\mathrm{x}^2(\mathrm{x}^2 - 1) < \mathrm{x}^3(\mathrm{x}^2 - 1)\), students often forget that \((\mathrm{x}^2 - 1)\) is negative when \(0 < \mathrm{x} < 1\). When dividing both sides by this negative term, they forget to flip the inequality sign, leading to an incorrect conclusion.

2. Arithmetic errors in power calculations

When verifying with concrete examples like x = 0.5, students frequently make calculation mistakes. For instance, computing \(\left(\frac{1}{2}\right)^5 = \frac{1}{32} = 0.03125\) or adding 0.0625 + 0.03125 = 0.09375. These arithmetic errors can make a correct statement appear false or vice versa, leading to wrong conclusions about which statements are true.

3. Incorrect term-by-term comparison in Statement II

For \(\mathrm{x}^4 + \mathrm{x}^5 < \mathrm{x}^3 + \mathrm{x}^2\), students might incorrectly try to compare \(\mathrm{x}^4\) with \(\mathrm{x}^3\) and \(\mathrm{x}^5\) with \(\mathrm{x}^2\), concluding that since \(\mathrm{x}^4 < \mathrm{x}^3\) but \(\mathrm{x}^5\) is not necessarily less than \(\mathrm{x}^2\), the inequality doesn't hold. The correct approach is to compare \(\mathrm{x}^4\) with \(\mathrm{x}^2\) and \(\mathrm{x}^5\) with \(\mathrm{x}^3\), then recognize that adding smaller terms to smaller terms yields a smaller sum.

Errors while selecting the answer

1. Misreading the answer choices

Students might correctly determine that all three statements are true but then select "D. I and II only" instead of "E. I, II and III" due to hasty reading. They might think they need to find which statements CAN be true rather than which statements MUST be true, or miss that option E includes all three statements.

2. Second-guessing correct reasoning

After correctly working through all three statements and finding them true, students might second-guess themselves thinking "this seems too easy" or "there must be a trick." They might then incorrectly conclude that one of the statements (often the more complex Statement III) must be false, leading them to choose "D. I and II only" instead of the correct answer E.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a strategic smart number
Since we need \(0 < \mathrm{x} < 1\), let's choose x = 0.5 (or \(\frac{1}{2}\)). This is an excellent choice because:

• It's exactly in the middle of our range, making it representative
• Powers of 0.5 are easy to calculate mentally
• It clearly demonstrates the behavior of fractions when raised to different powers

Step 2: Calculate all the powers we need
With x = 0.5:

• \(\mathrm{x}^2 = (0.5)^2 = 0.25\)
• \(\mathrm{x}^3 = (0.5)^3 = 0.125\)
• \(\mathrm{x}^4 = (0.5)^4 = 0.0625\)
• \(\mathrm{x}^5 = (0.5)^5 = 0.03125\)

Step 3: Test each inequality

Statement I: \(\mathrm{x}^5 < \mathrm{x}^3\)
0.03125 < 0.125 ✓ TRUE

Statement II: \(\mathrm{x}^4 + \mathrm{x}^5 < \mathrm{x}^3 + \mathrm{x}^2\)
Left side: 0.0625 + 0.03125 = 0.09375
Right side: 0.125 + 0.25 = 0.375
0.09375 < 0.375 ✓ TRUE

Statement III: \(\mathrm{x}^4 - \mathrm{x}^5 < \mathrm{x}^2 - \mathrm{x}^3\)
Left side: 0.0625 - 0.03125 = 0.03125
Right side: 0.25 - 0.125 = 0.125
0.03125 < 0.125 ✓ TRUE

Step 4: Verify the pattern holds generally
Our smart number x = 0.5 confirms that all three statements are true. The key insight is that for any fraction between 0 and 1, higher powers become progressively smaller. This makes x = 0.5 an excellent representative value that demonstrates the general behavior.

Answer: E (I, II, and III)