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Answer :
To solve the problem, we need to analyze the constraints given for the heights of Darius and his brother William. Here are the constraints described in the problem:
1. [tex]\( d \geq 36 \)[/tex]
2. [tex]\( w < 68 \)[/tex]
3. [tex]\( d \leq 4 + 2w \)[/tex]
Let's evaluate which statements must be true based on these constraints:
1. Darius is at least 36 inches tall.
This statement is derived directly from the first inequality: [tex]\( d \geq 36 \)[/tex]. This means Darius's height is indeed at least 36 inches. So, this statement must be true.
2. William's height is less than 68 inches.
The second inequality [tex]\( w < 68 \)[/tex] tells us that William's height is less than 68 inches. Therefore, this statement must also be true.
3. Darius is no more than 4 inches taller than twice William's height.
The third inequality [tex]\( d \leq 4 + 2w \)[/tex] indicates that Darius's height is no more than 4 inches added to twice William's height. This statement is a correct interpretation of the inequality and must be true.
Now, let's review the other possible statements to understand why they are not necessarily true:
- Darius is at most 36 inches tall.
This statement contradicts the first inequality [tex]\( d \geq 36 \)[/tex].
- William's height is at least 68 inches.
This contradicts the second inequality [tex]\( w < 68 \)[/tex].
- Darius is less than 4 inches taller than twice William's height.
The inequality [tex]\( d \leq 4 + 2w \)[/tex] implies that Darius is at most 4 inches taller, but not strictly less than, hence this statement does not have to be true.
In conclusion, the three statements that must be true about their heights are:
- Darius is at least 36 inches tall.
- William's height is less than 68 inches.
- Darius is no more than 4 inches taller than twice William's height.
1. [tex]\( d \geq 36 \)[/tex]
2. [tex]\( w < 68 \)[/tex]
3. [tex]\( d \leq 4 + 2w \)[/tex]
Let's evaluate which statements must be true based on these constraints:
1. Darius is at least 36 inches tall.
This statement is derived directly from the first inequality: [tex]\( d \geq 36 \)[/tex]. This means Darius's height is indeed at least 36 inches. So, this statement must be true.
2. William's height is less than 68 inches.
The second inequality [tex]\( w < 68 \)[/tex] tells us that William's height is less than 68 inches. Therefore, this statement must also be true.
3. Darius is no more than 4 inches taller than twice William's height.
The third inequality [tex]\( d \leq 4 + 2w \)[/tex] indicates that Darius's height is no more than 4 inches added to twice William's height. This statement is a correct interpretation of the inequality and must be true.
Now, let's review the other possible statements to understand why they are not necessarily true:
- Darius is at most 36 inches tall.
This statement contradicts the first inequality [tex]\( d \geq 36 \)[/tex].
- William's height is at least 68 inches.
This contradicts the second inequality [tex]\( w < 68 \)[/tex].
- Darius is less than 4 inches taller than twice William's height.
The inequality [tex]\( d \leq 4 + 2w \)[/tex] implies that Darius is at most 4 inches taller, but not strictly less than, hence this statement does not have to be true.
In conclusion, the three statements that must be true about their heights are:
- Darius is at least 36 inches tall.
- William's height is less than 68 inches.
- Darius is no more than 4 inches taller than twice William's height.
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