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Answer :
We are given the system of inequalities:
[tex]$$
\begin{aligned}
d &\geq 36, \\
w &< 68, \\
d &\leq 2w + 4.
\end{aligned}
$$[/tex]
Let's analyze each of the statements one by one:
1. "Darius is at least 36 inches tall."
This statement translates directly into the inequality
[tex]$$
d \geq 36.
$$[/tex]
Since this inequality is exactly what is given, it must be true.
2. "Darius is at most 36 inches tall."
The given inequality is [tex]$d \geq 36$[/tex], which means Darius is 36 inches or taller. Therefore, there is no guarantee that his height is at most 36 inches. This statement is not true.
3. "William's height is less than 68 inches."
This statement corresponds directly to
[tex]$$
w < 68.
$$[/tex]
Since this is one of the given inequalities, it must be true.
4. "William's height is at least 68 inches."
This statement would require that
[tex]$$
w \geq 68,
$$[/tex]
which is the opposite of the given [tex]$w < 68$[/tex]. Therefore, this statement is false.
5. "Darius is less than 4 inches taller than twice William's height."
To understand this statement, we first consider that twice William's height is [tex]$2w$[/tex]. Saying "Darius is less than 4 inches taller than twice William's height" means that the excess of Darius' height over [tex]$2w$[/tex] is at most 4 inches, or mathematically:
[tex]$$
d \leq 2w + 4.
$$[/tex]
This is exactly the third inequality in our system, so this statement must also be true.
Thus, the statements that must be true about their heights are:
- Statement 1: Darius is at least 36 inches tall.
- Statement 3: William's height is less than 68 inches.
- Statement 5: Darius is less than 4 inches taller than twice William's height.
The final answer is: Options 1, 3, and 5.
[tex]$$
\begin{aligned}
d &\geq 36, \\
w &< 68, \\
d &\leq 2w + 4.
\end{aligned}
$$[/tex]
Let's analyze each of the statements one by one:
1. "Darius is at least 36 inches tall."
This statement translates directly into the inequality
[tex]$$
d \geq 36.
$$[/tex]
Since this inequality is exactly what is given, it must be true.
2. "Darius is at most 36 inches tall."
The given inequality is [tex]$d \geq 36$[/tex], which means Darius is 36 inches or taller. Therefore, there is no guarantee that his height is at most 36 inches. This statement is not true.
3. "William's height is less than 68 inches."
This statement corresponds directly to
[tex]$$
w < 68.
$$[/tex]
Since this is one of the given inequalities, it must be true.
4. "William's height is at least 68 inches."
This statement would require that
[tex]$$
w \geq 68,
$$[/tex]
which is the opposite of the given [tex]$w < 68$[/tex]. Therefore, this statement is false.
5. "Darius is less than 4 inches taller than twice William's height."
To understand this statement, we first consider that twice William's height is [tex]$2w$[/tex]. Saying "Darius is less than 4 inches taller than twice William's height" means that the excess of Darius' height over [tex]$2w$[/tex] is at most 4 inches, or mathematically:
[tex]$$
d \leq 2w + 4.
$$[/tex]
This is exactly the third inequality in our system, so this statement must also be true.
Thus, the statements that must be true about their heights are:
- Statement 1: Darius is at least 36 inches tall.
- Statement 3: William's height is less than 68 inches.
- Statement 5: Darius is less than 4 inches taller than twice William's height.
The final answer is: Options 1, 3, and 5.
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