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3x+2y≤24; x≥5 & y≥2; then which of the following is a valid range?
- a)2≤y≤9/2
- b)5≤x≤20/3
- c)6≤x≤8
- d)More than one of the above
Correct answer is option 'D'. Can you explain this answer?
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3x+2y≤24; x≥5 & y≥2; then which of the following is a val...
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3x+2y≤24; x≥5 & y≥2; then which of the following is a val...
Understanding the Problem
We have the following inequalities to consider:
- 3x + 2y ≤ 24
- x ≥ 5
- y ≥ 2
We need to find the valid ranges for x and y based on these constraints.
Analyzing the Inequalities
1. Constraint from 3x + 2y ≤ 24:
- Rearranging gives us y ≤ (24 - 3x)/2.
2. Constraints on x and y:
- x must be at least 5.
- y must be at least 2.
Finding the Range for y
- Plugging x = 5 into the first inequality:
- y ≤ (24 - 3(5))/2 = (24 - 15)/2 = 9/2.
- Therefore, y can be in the range:
- 2 ≤ y ≤ 9/2.
Finding the Range for x
- We know x must be at least 5.
- To find the upper limit for x, we can set y at its minimum value (y = 2):
- 3x + 2(2) ≤ 24
- 3x + 4 ≤ 24
- 3x ≤ 20
- x ≤ 20/3.
- Therefore, the range for x is:
- 5 ≤ x ≤ 20/3.
Evaluating the Options
- a) 2 ≤ y ≤ 9/2: Valid based on our analysis.
- b) 5 ≤ x ≤ 20/3: Also valid.
- c) 6 ≤ x ≤ 8: This is a subset of the valid x range but not fully valid since x can also be 5 and 20/3.
- d) More than one of the above: This is correct as both (a) and (b) are valid.
Conclusion
The correct answer is indeed option 'D' as both ranges a) and b) are valid.
We have the following inequalities to consider:
- 3x + 2y ≤ 24
- x ≥ 5
- y ≥ 2
We need to find the valid ranges for x and y based on these constraints.
Analyzing the Inequalities
1. Constraint from 3x + 2y ≤ 24:
- Rearranging gives us y ≤ (24 - 3x)/2.
2. Constraints on x and y:
- x must be at least 5.
- y must be at least 2.
Finding the Range for y
- Plugging x = 5 into the first inequality:
- y ≤ (24 - 3(5))/2 = (24 - 15)/2 = 9/2.
- Therefore, y can be in the range:
- 2 ≤ y ≤ 9/2.
Finding the Range for x
- We know x must be at least 5.
- To find the upper limit for x, we can set y at its minimum value (y = 2):
- 3x + 2(2) ≤ 24
- 3x + 4 ≤ 24
- 3x ≤ 20
- x ≤ 20/3.
- Therefore, the range for x is:
- 5 ≤ x ≤ 20/3.
Evaluating the Options
- a) 2 ≤ y ≤ 9/2: Valid based on our analysis.
- b) 5 ≤ x ≤ 20/3: Also valid.
- c) 6 ≤ x ≤ 8: This is a subset of the valid x range but not fully valid since x can also be 5 and 20/3.
- d) More than one of the above: This is correct as both (a) and (b) are valid.
Conclusion
The correct answer is indeed option 'D' as both ranges a) and b) are valid.
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3x+2y≤24; x≥5 & y≥2; then which of the following is a valid range?a)2≤y≤9/2b)5≤x≤20/3c)6≤x≤8d)More than one of the aboveCorrect answer is option 'D'. Can you explain this answer?
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