To solve the problem, we need to determine the inequality that represents all possible combinations of penny nails ([tex]\(x\)[/tex]) and wood nails ([tex]\(y\)[/tex]) that can fit into a 1-pound bag.
1. Understand the problem:
- Each penny nail weighs 0.05 pounds.
- Each wood nail weighs 0.1 pounds.
- The total weight capacity of the bag is 1 pound.
2. Formulate the weight constraint:
The total weight of [tex]\(x\)[/tex] penny nails and [tex]\(y\)[/tex] wood nails should not exceed 1 pound. Therefore, we can write this constraint as:
[tex]\[
0.05x + 0.1y \leq 1
\][/tex]
3. Simplify the inequality:
To make it easier to understand and possibly recognize the correct inequality form, we can eliminate the decimals by multiplying the entire inequality by 10:
[tex]\[
10(0.05x + 0.1y) \leq 10 \times 1
\][/tex]
Simplifying this, we get:
[tex]\[
0.5x + y \leq 10
\][/tex]
4. Express [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
To match one of the options given, we isolate [tex]\(y\)[/tex]:
[tex]\[
y \leq 10 - 0.5x
\][/tex]
Rearranging this to match the given format in the problem, we get:
[tex]\[
y \leq -0.5x + 10
\][/tex]
5. Compare with the given options:
Now, we compare our resulting inequality with the options provided:
- A. [tex]\(y \geq -\frac{1}{2}x + 10\)[/tex]
- B. [tex]\(\frac{1}{10}x + \frac{1}{20}y \geq 1\)[/tex]
- C. [tex]\(y \leq -\frac{1}{2}x + 10\)[/tex]
- D. [tex]\(\frac{1}{20}x + \frac{1}{10}y \geq 1\)[/tex]
The inequality we derived is [tex]\(y \leq -0.5x + 10\)[/tex], which matches option C.
Therefore, the correct inequality is:
[tex]\(
\boxed{y \leq -\frac{1}{2}x + 10}
\)[/tex]
This corresponds to option C.
