Question

warm-up

amad said, \\\(3n\\) is always greater than \\(n + 3\\).\ do you agree with him?

• yes, because multiplication always gives a larger answer than addition.
• yes, because \\(n\\) must be a positive number.
• no, because multiplication is not the opposite of addition.
• no, because \\(3n\\) could be equal to or less than \\(n + 3\\).

Explanation:

Analyze the statement

The statement claims that \(3n\) is always greater than \(n + 3\) for any value of \(n\). To test this claim, we set up the inequality:

$$3n > n + 3$$

Solve the inequality

We solve the inequality to find the range of values for which the statement is true:

$$
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$$

This shows that the statement is only true when \(n > 1.5\).

Find counterexamples

If \(n \le 1.5\), the statement is false. Let's test a few values:

• If \(n = 1\): \(3(1) = 3\) and \(1 + 3 = 4\). Here, \(3 < 4\), so \(3n\) is less than \(n + 3\).
• If \(n = 0\): \(3(0) = 0\) and \(0 + 3 = 3\). Here, \(0 < 3\), so \(3n\) is less than \(n + 3\).

Evaluate the options

• "Yes, because multiplication always gives a larger answer than addition." (Incorrect, as shown by counterexamples).
• "Yes, because n must be a positive number." (Incorrect, even for positive \(n = 1\), the statement is false).
• "No, because multiplication is not the opposite of addition." (Incorrect reasoning).
• "No, because \(3n\) could be equal to or less than \(n + 3\)." (Correct, since \(3n \le n + 3\) when \(n \le 1.5\)).

Answer:

• Yes, because multiplication always gives a larger answer than addition.
• Yes, because n must be a positive number.
• No, because multiplication is not the opposite of addition.
• No, because \(3n\) could be equal to or less than \(n + 3\). (Correct answer)