Answer :
Let's analyze and solve the given system of inequalities step-by-step to determine which statements must be true about the heights of Darius, [tex]\( d \)[/tex], and his brother William, [tex]\( w \)[/tex].
The system of inequalities is:
1. [tex]\( d \geq 36 \)[/tex]
2. [tex]\( w < 68 \)[/tex]
3. [tex]\( d \leq 4 + 2w \)[/tex]
Based on these inequalities, we can analyze each statement individually:
1. Darius is at least 36 inches tall.
This statement corresponds to the inequality [tex]\( d \geq 36 \)[/tex].
Since this directly matches the first inequality in the system, it must be true.
Thus, this statement is true.
2. Darius is at most 36 inches tall.
This statement implies that [tex]\( d \leq 36 \)[/tex].
However, the first inequality in the system already states [tex]\( d \geq 36 \)[/tex], which contradicts [tex]\( d \leq 36 \)[/tex]. Hence, Darius cannot be at most 36 inches tall.
Thus, this statement is false.
3. William's height is less than 68 inches.
This statement corresponds to the inequality [tex]\( w < 68 \)[/tex].
Since this directly matches the second inequality in the system, it must be true.
Thus, this statement is true.
4. William's height is at least 68 inches.
This statement implies that [tex]\( w \geq 68 \)[/tex].
However, the second inequality in the system already states [tex]\( w < 68 \)[/tex], which contradicts [tex]\( w \geq 68 \)[/tex]. Hence, William cannot be at least 68 inches tall.
Thus, this statement is false.
5. Darius is less than 4 inches taller than twice William's height.
This statement implies [tex]\( d < 2w + 4 \)[/tex].
However, the third inequality in the system states [tex]\( d \leq 2w + 4 \)[/tex]. The statement "less than" (which implies [tex]\( < \)[/tex]) is stricter than "less than or equal to" (which implies [tex]\( \leq \)[/tex]), but the third inequality does not guarantee that [tex]\( d \)[/tex] is strictly less than [tex]\( 2w + 4 \)[/tex].
Thus, this statement is false.
6. Darius is no more than 4 inches taller than twice William's height.
This statement corresponds to the inequality [tex]\( d \leq 2w + 4 \)[/tex].
Since this directly matches the third inequality in the system, it must be true.
Thus, this statement is true.
Therefore, the three true statements about their heights are:
1. Darius is at least 36 inches tall.
2. William's height is less than 68 inches.
3. Darius is no more than 4 inches taller than twice William's height.
The system of inequalities is:
1. [tex]\( d \geq 36 \)[/tex]
2. [tex]\( w < 68 \)[/tex]
3. [tex]\( d \leq 4 + 2w \)[/tex]
Based on these inequalities, we can analyze each statement individually:
1. Darius is at least 36 inches tall.
This statement corresponds to the inequality [tex]\( d \geq 36 \)[/tex].
Since this directly matches the first inequality in the system, it must be true.
Thus, this statement is true.
2. Darius is at most 36 inches tall.
This statement implies that [tex]\( d \leq 36 \)[/tex].
However, the first inequality in the system already states [tex]\( d \geq 36 \)[/tex], which contradicts [tex]\( d \leq 36 \)[/tex]. Hence, Darius cannot be at most 36 inches tall.
Thus, this statement is false.
3. William's height is less than 68 inches.
This statement corresponds to the inequality [tex]\( w < 68 \)[/tex].
Since this directly matches the second inequality in the system, it must be true.
Thus, this statement is true.
4. William's height is at least 68 inches.
This statement implies that [tex]\( w \geq 68 \)[/tex].
However, the second inequality in the system already states [tex]\( w < 68 \)[/tex], which contradicts [tex]\( w \geq 68 \)[/tex]. Hence, William cannot be at least 68 inches tall.
Thus, this statement is false.
5. Darius is less than 4 inches taller than twice William's height.
This statement implies [tex]\( d < 2w + 4 \)[/tex].
However, the third inequality in the system states [tex]\( d \leq 2w + 4 \)[/tex]. The statement "less than" (which implies [tex]\( < \)[/tex]) is stricter than "less than or equal to" (which implies [tex]\( \leq \)[/tex]), but the third inequality does not guarantee that [tex]\( d \)[/tex] is strictly less than [tex]\( 2w + 4 \)[/tex].
Thus, this statement is false.
6. Darius is no more than 4 inches taller than twice William's height.
This statement corresponds to the inequality [tex]\( d \leq 2w + 4 \)[/tex].
Since this directly matches the third inequality in the system, it must be true.
Thus, this statement is true.
Therefore, the three true statements about their heights are:
1. Darius is at least 36 inches tall.
2. William's height is less than 68 inches.
3. Darius is no more than 4 inches taller than twice William's height.
