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Answer :
To solve the problem and determine which statements are true about the heights of Darius and William, let's examine each inequality and statement carefully:
1. Inequality [tex]$d \geq 36$[/tex]:
- This means that Darius must be at least 36 inches tall.
- Therefore, the statement "Darius is at least 36 inches tall" is true.
2. Inequality [tex]$w < 68$[/tex]:
- This indicates that William's height must be less than 68 inches.
- So, the statement "William's height is less than 68 inches" is true.
3. Inequality [tex]$d \leq 4 + 2w$[/tex]:
- This means the height of Darius must not exceed 4 inches more than twice William's height.
- The statement "Darius is no more than 4 inches taller than twice William's height" is true.
Now, let's evaluate the statements given:
- Statement 1: "Darius is at least 36 inches tall."
- This statement is consistent with the inequality [tex]$d \geq 36$[/tex] and is therefore true.
- Statement 2: "Darius is at most 36 inches tall."
- This is incorrect because Darius's height can be more than 36 inches.
- Statement 3: "William's height is less than 68 inches."
- This matches the inequality [tex]$w < 68$[/tex] and is true.
- Statement 4: "William's height is at least 68 inches."
- This contradicts [tex]$w < 68$[/tex], so it is false.
- Statement 5: "Darius is less than 4 inches taller than twice William's height."
- This does not align with the expression, as it suggests Darius is strictly less than [tex]$4 + 2w$[/tex], which isn't guaranteed.
- Statement 6: "Darius is no more than 4 inches taller than twice William's height."
- This is accurate because of the inequality [tex]$d \leq 4 + 2w$[/tex].
Therefore, the true statements 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. Inequality [tex]$d \geq 36$[/tex]:
- This means that Darius must be at least 36 inches tall.
- Therefore, the statement "Darius is at least 36 inches tall" is true.
2. Inequality [tex]$w < 68$[/tex]:
- This indicates that William's height must be less than 68 inches.
- So, the statement "William's height is less than 68 inches" is true.
3. Inequality [tex]$d \leq 4 + 2w$[/tex]:
- This means the height of Darius must not exceed 4 inches more than twice William's height.
- The statement "Darius is no more than 4 inches taller than twice William's height" is true.
Now, let's evaluate the statements given:
- Statement 1: "Darius is at least 36 inches tall."
- This statement is consistent with the inequality [tex]$d \geq 36$[/tex] and is therefore true.
- Statement 2: "Darius is at most 36 inches tall."
- This is incorrect because Darius's height can be more than 36 inches.
- Statement 3: "William's height is less than 68 inches."
- This matches the inequality [tex]$w < 68$[/tex] and is true.
- Statement 4: "William's height is at least 68 inches."
- This contradicts [tex]$w < 68$[/tex], so it is false.
- Statement 5: "Darius is less than 4 inches taller than twice William's height."
- This does not align with the expression, as it suggests Darius is strictly less than [tex]$4 + 2w$[/tex], which isn't guaranteed.
- Statement 6: "Darius is no more than 4 inches taller than twice William's height."
- This is accurate because of the inequality [tex]$d \leq 4 + 2w$[/tex].
Therefore, the true statements 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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