Let's break down the problem statement step-by-step to identify the correct inequalities that model this situation:
1. Budget Constraint:
Adam can spend up to [tex]$150 on his project. Black stone tiles cost $[/tex]5 each and gray stone tiles cost [tex]$4 each. The total cost can be represented by the inequality:
\[
5x + 4y \leq 150
\]
Additionally, there can also be a strict inequality if we consider the possibility of spending strictly less than $[/tex]150:
[tex]\[
5x + 4y < 150
\][/tex]
2. Gray Tiles Constraint:
Adam wants the number of gray stone tiles to be less than half the number of black stone tiles. This can be represented as:
[tex]\[
y < \frac{1}{2}x
\][/tex]
3. Non-negative Constraint for Black Stone Tiles:
The number of black stone tiles, [tex]\(x\)[/tex], must be non-negative. Therefore, we have:
[tex]\[
x \geq 0
\][/tex]
4. Positive Number Constraint (Considering x must be positive):
Sometimes, it's important to specify that the number of tiles must be strictly positive:
[tex]\[
x > 0
\][/tex]
5. Extra constraint which doesn't directly apply (Incorrect):
[tex]\[
y \leq \frac{1}{2}
\][/tex]
This inequality is incorrect in this context because it doesn't represent the relationship between the number of gray and black stone tiles.
Now, listing the valid inequalities that model Adam's situation:
- [tex]\( x > 0 \)[/tex]
- [tex]\( x \geq 0 \)[/tex]
- [tex]\( y < \frac{1}{2} x \)[/tex]
- [tex]\( 5x + 4y \leq 150 \)[/tex]
- [tex]\( 5x + 4y < 150 \)[/tex]
So the correct inequalities are:
- [tex]\( x > 0 \)[/tex]
- [tex]\( x \geq 0 \)[/tex]
- [tex]\( y < \frac{1}{2} x \)[/tex]
- [tex]\( 5x + 4y \leq 150 \)[/tex]
- [tex]\( 5x + 4y < 150 \)[/tex]
