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Answer :
To solve the problem, let's break down the given system of inequalities and identify what statements must be true:
1. Inequality 1: [tex]\( d \geq 36 \)[/tex]
This inequality tells us that Darius is at least 36 inches tall. Therefore, it is true that Darius is at least 36 inches tall.
2. Inequality 2: [tex]\( w < 68 \)[/tex]
This inequality tells us that William's height is less than 68 inches. Therefore, it is true that William's height is less than 68 inches.
3. Inequality 3: [tex]\( d \leq 4 + 2w \)[/tex]
This inequality places a restriction on Darius's height related to William's height. It means Darius's height must be no more than 4 inches plus twice William's height.
Let's explore a bit further:
- Combining Inequalities 1 ([tex]\( d \geq 36 \)[/tex]) and 3 ([tex]\( d \leq 4 + 2w \)[/tex]), we consider a value range for [tex]\( w \)[/tex].
- To find the minimum value for [tex]\( w \)[/tex] when [tex]\( d = 36 \)[/tex], we solve:
[tex]\[
36 \leq 4 + 2w
\][/tex]
Subtract 4 from both sides:
[tex]\[
32 \leq 2w
\][/tex]
Divide by 2:
[tex]\[
16 \leq w
\][/tex]
This means that for [tex]\( d \)[/tex] to be at least 36, [tex]\( w \)[/tex] must be at least 16 inches tall, but still less than 68 inches.
Based on these inequalities, we can confirm the following statements as true:
- 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, which follows from the inequality [tex]\( d \leq 4 + 2w \)[/tex].
These are the statements supported by the given system of linear inequalities.
1. Inequality 1: [tex]\( d \geq 36 \)[/tex]
This inequality tells us that Darius is at least 36 inches tall. Therefore, it is true that Darius is at least 36 inches tall.
2. Inequality 2: [tex]\( w < 68 \)[/tex]
This inequality tells us that William's height is less than 68 inches. Therefore, it is true that William's height is less than 68 inches.
3. Inequality 3: [tex]\( d \leq 4 + 2w \)[/tex]
This inequality places a restriction on Darius's height related to William's height. It means Darius's height must be no more than 4 inches plus twice William's height.
Let's explore a bit further:
- Combining Inequalities 1 ([tex]\( d \geq 36 \)[/tex]) and 3 ([tex]\( d \leq 4 + 2w \)[/tex]), we consider a value range for [tex]\( w \)[/tex].
- To find the minimum value for [tex]\( w \)[/tex] when [tex]\( d = 36 \)[/tex], we solve:
[tex]\[
36 \leq 4 + 2w
\][/tex]
Subtract 4 from both sides:
[tex]\[
32 \leq 2w
\][/tex]
Divide by 2:
[tex]\[
16 \leq w
\][/tex]
This means that for [tex]\( d \)[/tex] to be at least 36, [tex]\( w \)[/tex] must be at least 16 inches tall, but still less than 68 inches.
Based on these inequalities, we can confirm the following statements as true:
- 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, which follows from the inequality [tex]\( d \leq 4 + 2w \)[/tex].
These are the statements supported by the given system of linear inequalities.
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