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.\
\\( \square \\) darius is at least 36 inches tall.\
\\( \square \\) darius is at most 36 inches tall.\
\\( \square \\) williams height is less than 68 inches.\
\\( \square \\) william’s height is at least 68 inches.\
\\( \square \\) darius is less than 4 inches taller than twice williams height.\
\\( \square \\) darius is no more than 4 inches taller than twice williams height.

Explanation:

1. Analyze \( d \geq 36 \): This means Darius's height \( d \) is greater than or equal to 36 inches, so "Darius is at least 36 inches tall" is true.
2. Analyze \( w < 68 \): This means William's height \( w \) is less than 68 inches, so "William's height is less than 68 inches" is true.
3. Analyze \( d \leq 4 + 2w \): Rearranging, \( d - 2w \leq 4 \), which means 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" is true.

• "Darius is at most 36 inches tall" is false (contradicts \( d \geq 36 \)).
• "William’s height is at least 68 inches" is false (contradicts \( w < 68 \)).
• "Darius is less than 4 inches taller than twice William’s height" is false (the inequality is \( \leq \), not \( < \)).

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.