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Answer :
We start with the given system of inequalities for Darius’s height [tex]$d$[/tex] and Whiliam’s height [tex]$w$[/tex]:
[tex]$$
\begin{aligned}
d &\ge 36, \\
w &< 68, \\
d &\le 4 + 2w.
\end{aligned}
$$[/tex]
Let’s analyze each statement:
1. The inequality
[tex]$$
d \ge 36
$$[/tex]
tells us that Darius must be at least 36 inches tall.
2. The inequality
[tex]$$
w < 68
$$[/tex]
implies that Whiliam’s height is less than 68 inches.
3. The third inequality
[tex]$$
d \le 4 + 2w
$$[/tex]
can be interpreted as a relation between Darius's and Whiliam's heights. It states that Darius's height is no more than 4 inches added to twice Whiliam’s height. In other words, Darius is no more than 4 inches shorter than twice Whiliam’s height.
Therefore, the three statements that must be true about their heights are:
- Darius is at least 36 inches tall.
- Whiliam’s height is less than 68 inches.
- Darius is no more than 4 inches shorter than twice Whiliam’s height.
This corresponds to the options:
[tex]$$\boxed{1, \; 3, \; 5.}$$[/tex]
[tex]$$
\begin{aligned}
d &\ge 36, \\
w &< 68, \\
d &\le 4 + 2w.
\end{aligned}
$$[/tex]
Let’s analyze each statement:
1. The inequality
[tex]$$
d \ge 36
$$[/tex]
tells us that Darius must be at least 36 inches tall.
2. The inequality
[tex]$$
w < 68
$$[/tex]
implies that Whiliam’s height is less than 68 inches.
3. The third inequality
[tex]$$
d \le 4 + 2w
$$[/tex]
can be interpreted as a relation between Darius's and Whiliam's heights. It states that Darius's height is no more than 4 inches added to twice Whiliam’s height. In other words, Darius is no more than 4 inches shorter than twice Whiliam’s height.
Therefore, the three statements that must be true about their heights are:
- Darius is at least 36 inches tall.
- Whiliam’s height is less than 68 inches.
- Darius is no more than 4 inches shorter than twice Whiliam’s height.
This corresponds to the options:
[tex]$$\boxed{1, \; 3, \; 5.}$$[/tex]
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