To express the constraints given in the problem in the form of linear inequalities, we need to consider the information carefully:
1. Storage Space Constraint:
- The manufacturer has storage space for a maximum of 100 items.
- Let [tex]\( x \)[/tex] be the number of tables.
- Let [tex]\( y \)[/tex] be the number of chairs.
- Therefore, the inequality representing this constraint is:
[tex]\[
x + y \leq 100
\][/tex]
2. Investment Constraint:
- The manufacturer has [tex]$\$[/tex]10,000[tex]$ to invest.
- Each table costs \$[/tex]400.
- Each chair costs \$100.
- Therefore, the total cost for [tex]\( x \)[/tex] tables and [tex]\( y \)[/tex] chairs would be [tex]\( 400x + 100y \)[/tex].
- The inequality representing this constraint is:
[tex]\[
400x + 100y \leq 10000
\][/tex]
- This can be simplified by dividing by 100:
[tex]\[
4x + y \leq 100
\][/tex]
3. Non-Negative Constraints:
- The number of tables and chairs cannot be negative.
- Therefore, the inequalities are:
[tex]\[
x \geq 0
\][/tex]
[tex]\[
y \geq 0
\][/tex]
Combining all these constraints, we get the system of linear inequalities as:
[tex]\[
\begin{cases}
x + y \leq 100 \\
4x + y \leq 100 \\
x \geq 0 \\
y \geq 0
\end{cases}
\][/tex]
Looking at the given options, we see that:
(a) [tex]\( x + y \leq 100, 4x + y \leq 100, x \geq 0, y \geq 0 \)[/tex]
This matches our derived system of inequalities exactly. Therefore, the correct answer is:
(a) [tex]\( x + y \leq 100, 4x + y \leq 100, x \geq 0, y \geq 0 \)[/tex]
