Question

this system of linear inequalities can be used to find the possible heights, in inches, of darius, d, and his brother william, w:
( d geq 36 )
( w < 68 )
( d leq 4 + 2w )
which statements must be true about their heights?
select three options.
☐ darius is at least 36 inches tall.
☐ darius is at most 36 inches tall.
☐ william’s height is less than 68 inches.
☐ william’s height is at least 68 inches.
☐ darius is less than 4 inches taller than twice william’s height.
☐ darius is no more than 4 inches taller than twice william’s height.

Explanation:

Step1: Analyze \( d \geq 36 \)

The inequality \( d \geq 36 \) means Darius's height \( d \) is greater than or equal to 36 inches, so "Darius is at least 36 inches tall" is true. "Darius is at most 36 inches tall" (which would be \( d \leq 36 \)) is false.

Step2: Analyze \( w < 68 \)

The inequality \( w < 68 \) means William's height \( w \) is less than 68 inches, so "William’s height is less than 68 inches" is true. "William’s height is at least 68 inches" (which would be \( w \geq 68 \)) is false.

Step3: Analyze \( d \leq 4 + 2w \)

The inequality \( d \leq 4 + 2w \) can be rewritten as \( d - 2w \leq 4 \), meaning Darius's height minus twice William's height is at most 4 inches. So Darius is no more than 4 inches taller than twice William’s height (since \( d \leq 4 + 2w \) implies \( d - 2w \leq 4 \), or \( d \leq 2w + 4 \), so the difference \( d - 2w \) is at most 4). "Darius is less than 4 inches taller than twice William’s height" is not necessarily true (it could be equal to 4), but "Darius is no more than 4 inches taller than twice William’s height" is true.

Answer:

• 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.