Question

A student needs to buy notebooks (\(n\)) for a semester. Double the number of notebooks plus \(5\) must strictly exceed \(15\), but the number of notebooks plus \(10\) must be no more than \(22\). What is the range of notebooks they can buy?

• A. \(\{5,6,7,8,9,10,11,12\}\)
• B. \(\{6,7,8,9,10,11\}\)
• C. \(\{6,7,8,9,10,11,12\}\)
• D. \(\{5,6,7,8,9,10,11,12,13,14,15\}\)

Hint:

For simultaneous inequalities, solve each inequality separately and then find the common interval satisfying both conditions.

Answer 1

The Correct Option is C

To solve this problem, we need to determine the number of notebooks (\(\(n\)\)) that meet the specified conditions. We are given two inequalities to work with:

1. The first condition states that double the number of notebooks plus 5 must strictly exceed 15. This can be translated into the inequality: \(2n + 5 > 15\).
2. The second condition states that the number of notebooks plus 10 must be no more than 22. This is translated into another inequality: \(n + 10 \leq 22\).

Let us solve each inequality separately:

1. Starting with the first inequality:
   • \(2n + 5 > 15\)
   • Subtract 5 from both sides: \(2n > 10\)
   • Divide by 2: \(n > 5\)
   • This inequality tells us that \(n\), the number of notebooks, must be greater than 5.
2. Now, solve the second inequality:
   • \(n + 10 \leq 22\)
   • Subtract 10 from both sides: \(n \leq 12\)
   • This inequality states that \(n\) must be no greater than 12.

Combining the solutions from these two inequalities, we get:

• \(5 < n \leq 12\)

This means that the number of notebooks (\(n\)) that the student can buy must be greater than 5 and up to 12. Therefore, the possible values for \(n\) are 6, 7, 8, 9, 10, 11, and 12.

Hence, the correct option is \(\{6,7,8,9,10,11,12\}\).