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A manufacturer produces two items, A and B. He has [tex]$10,000 to invest and space to store 100 items. A table costs him $[/tex]400 and a chair $100. Express this in the form of linear inequalities.

A. [tex]\( x + y \leq 100 \)[/tex], [tex]\( 4x + y \leq 10,000 \)[/tex], [tex]\( x \geq 0 \)[/tex], [tex]\( y \geq 0 \)[/tex]
B. [tex]\( x + y \leq 1000 \)[/tex], [tex]\( 2x + 5y \ \textless \ 1000 \)[/tex], [tex]\( x \geq 0 \)[/tex], [tex]\( y \geq 0 \)[/tex]
C. [tex]\( x + y \ \textgreater \ 100 \)[/tex], [tex]\( 4x + y \geq 10,000 \)[/tex], [tex]\( x \geq 0 \)[/tex], [tex]\( y \geq 0 \)[/tex]
D. None of these

Sagot :

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]
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