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23 Nov 2025, 23:16
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A
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66% (01:18) correct
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If −5 ≤ a ≤ 12 and −10 ≤ b ≤ 25, which of the following represents all possible values of a – b ?
A. −30 ≤ a – b ≤ 22
B. −30 ≤ a – b ≤ 5
C. −13 ≤ a – b ≤ 5
D. 5 ≤ a – b ≤ 22
E. 5 ≤ a – b ≤ 30
A. −30 ≤ a – b ≤ 22
B. −30 ≤ a – b ≤ 5
C. −13 ≤ a – b ≤ 5
D. 5 ≤ a – b ≤ 22
E. 5 ≤ a – b ≤ 30
24 Nov 2025, 01:51
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Classic min max problem.
Min a-b by doing:
- Minimize value of a, as a is positive in the equation "a-b". Therefore, a = -5
- Maximize value of b, as it is being subtracted. Therefore, b = 25
Equation: a-b = -5 - 25 = -30
Max a-b by doing:
- Max value of a -> Therefore a = 12
- Minimize b, because any negative value of b will be added to the output of the formula -> -10
12 - (-10) = 22
ANS: A
Min a-b by doing:
- Minimize value of a, as a is positive in the equation "a-b". Therefore, a = -5
- Maximize value of b, as it is being subtracted. Therefore, b = 25
Equation: a-b = -5 - 25 = -30
Max a-b by doing:
- Max value of a -> Therefore a = 12
- Minimize b, because any negative value of b will be added to the output of the formula -> -10
12 - (-10) = 22
ANS: A
24 Nov 2025, 01:58
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Bunuel
To find the range of possible values for \(a - b\), we need to determine the minimum and maximum possible values for this expression.
CRITICAL RULE: You cannot subtract inequalities directly (i.e., you cannot just do "top minus top" and "bottom minus bottom"). Doing so often leads to errors.
[hr]
Method 1: Minimize and Maximize (Logical Approach)
To find the range, ask yourself two questions:
1. What is the smallest possible value (Minimum)?
To make \(a - b\) as small as possible, you want the smallest possible \(a\) and subtract the largest possible amount (\(b\)).
\(Min(a - b) = Min(a) - Max(b)\)
\(Min(a - b) = -5 - 25 = -30\)
2. What is the largest possible value (Maximum)?
To make \(a - b\) as large as possible, you want the largest possible \(a\) and subtract the smallest possible amount (\(b\)).
\(Max(a - b) = Max(a) - Min(b)\)
\(Max(a - b) = 12 - (-10)\)
\(Max(a - b) = 12 + 10 = 22\)
So, the range is \(-30 \le a - b \le 22\).
[hr]
Method 2: Algebraic Addition
It is safer to turn subtraction into addition: \(a - b\) is the same as \(a + (-b)\).
1. Start with the range of \(b\):
\(-10 \le b \le 25\)
2. Multiply the entire inequality by \(-1\) to find the range of \(-b\).
(Remember: When multiplying by a negative number, flip the inequality signs or swap the limits).
\(-25 \le -b \le 10\)
3. Now, add this inequality to the range of \(a\):
\(\,\,\,\,\, -5 \le a \le 12\)
+ -25 \le -b \le 10
\( -30 \le a - b \le 22\)
This matches Option (A).
Answer: A
