1. Look at the cards that we know: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
2. There are two cards in each envelope, and the two cards combined equal the number on the envelope
3. The envelopes have the sums noted: 3, 7, 8, 13, 14
4. In the "8" envelope, we need to figure out what two cards will add up to 8
5. List the potential pairs which sum to 8:
0 + 8
1 + 7
2 + 6
3 + 5
4 + 4 (but there is only one "4" card)
6. Ruling out some pairs from the other envelope totals is possible
7. The potential pairs for the "8" envelope are:
0 and 8
1 and 7
2 and 6
Therefore, the potential pairs of cards in the "8" envelope are:
•0 and 8
•1 and 7
•2 and 6
• 3 and 5
Both pairs would satisfy the condition of summing up to 8.
Problem 1: Envelope and Card Combinations
Known Cards: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Envelope Rules:
There are two cards in every envelope.
The envelope sums are: 3, 7, 8, 13, 14
Task: Determine the possible pairs of cards in the envelope with "8".
Step 1: List Possible Pairs Summing to 8
Possible pairs (no duplicates allowed):
0 + 8
1 + 7
2 + 6
3 + 5
4 + 4 (impossible, there is only one "4" card)
Step 2: Cross Out Impossible Pairs
Considering other envelope sums, good pairs for "8" are:
0 and 8
1 and 7
2 and 6
3 and 5
Problem 2: Solving the Inequality n+10>2n+3n+10>2n+3
Step 1: Write the Inequality
Two amounts are being compared:
nn+10
2n+32n+3
Objective: Find values of nn where n+10>2n+3n+10>2n+3.
Step 2: Solve Algebraically
Subtract 2n2n from both sides:
n+10−2n>3n+10−2n>3
−n+10>3
−n+10>3
Subtract 10 from both sides:
−n>−7
−n>−7
Multiply by −1−1 (reverse inequality sign):
n<7
Important Note: Dividing/multiplying by negative number reverses the inequality.
Step 3: Check the Solution
For n<7n<7: n+10>2n+3n+10>2n+3 (e.g., n=5n=5: 15>1315>13).
For n>7n>7: 2n+3>n+102n+3>n+10 (e.g., n=8n=8: 19>1819>18).
At n=7n=7: Both equal 17.
Graphical Interpretation
The lines y=n+10y=n+10 and y=2n+3y=2n+3 cross at n=7n=7.
n+10n+10 lies above 2n+32n+3 when n<7n<7.
General Strategy for Comparing Functions
Write f(n)>g(n)f(n)>g(n).
Rearrange to solve for nn.
Solve, obeying inequality rules.
Calculate the solution to determine dominance regions.