To determine the correct inequality that represents all combinations of penny nails [tex]\( x \)[/tex] and wood nails [tex]\( y \)[/tex] that can fit within a 1-pound bag, let's break it down step-by-step.
1. Identify the problem: We need to establish an inequality that ensures the total weight of the nails in the bag does not exceed 1 pound.
2. Define the weights:
- Each penny nail weighs 0.05 pounds.
- Each wood nail weighs 0.1 pounds.
- The maximum weight the bag can hold is 1 pound.
3. Set up the inequality: We know that the combined weight of [tex]\( x \)[/tex] penny nails and [tex]\( y \)[/tex] wood nails must be less than or equal to 1 pound. Mathematically, this can be written as:
[tex]\[
0.05x + 0.1y \leq 1
\][/tex]
4. Simplify the form of the inequality (optional but useful for clarity): This inequality can be written directly without further simplification for checking against the provided options.
Now let's compare this to the given options:
A. [tex]\( y \geq -\frac{1}{2}x + 10 \)[/tex]
- This option suggests a linear relationship but is written as an inequality where [tex]\( y \)[/tex] is greater than or equal to a term involving [tex]\( x \)[/tex]. This does not match our requirement and might represent a boundary condition rather than the weight limit.
B. [tex]\( \frac{1}{10} x + \frac{1}{20} y \geq 1 \)[/tex]
- This option presents fractions as coefficients, which are reciprocals of what we have. It does not correspond to our inequality [tex]\( 0.05x + 0.1y \leq 1 \)[/tex].
C. [tex]\( y \leq -\frac{1}{2} x + 10 \)[/tex]
- This option also suggests a linear relationship but with [tex]\( y \)[/tex] being less than or equal to a term involving [tex]\( x \)[/tex]. It reflects the boundary condition of the weight in terms of [tex]\( y \)[/tex] as dependent on [tex]\( x \)[/tex].
D. [tex]\( \frac{1}{20} x + \frac{1}{10} y \geq 1 \)[/tex]
- This option again changes the coefficients in a way that does not match our requirement.
Comparing carefully, we align our obtained inequality [tex]\( 0.05x + 0.1y \leq 1 \)[/tex] with:
[tex]\[ y \leq -\frac{1}{2} x + 10 \][/tex]
Reviewing these steps clearly directs us that the correct answer is:
C. [tex]\( y \leq -\frac{1}{2} x + 10 \)[/tex]
