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x cannot be equal to -5

Given the equation |x| + |y| = 4, the possible values of x and y must satisfy this condition while also ensuring that x is not equal to y. The value -5 would imply that |x| = 5, which cannot be true since that would require |y| to be -1, an impossible scenario.

A) 2

If x = 2, then |x| + |y| = 4 implies |y| = 2. In this case, y could equal 2 or -2. Since x ≠ y, it is possible for x to equal 2 while satisfying the equation.

B)

This option is blank and does not provide any relevant information to assess.

C) -2

If x = -2, then |x| + |y| = 4 implies |y| = 6. This means y could be 6 or -6, which allows for the condition that x ≠ y to hold true. Thus, x can equal -2.

D) -5

Setting x = -5 gives |x| = 5. In this case, |y| must equal -1, which is not possible since absolute values cannot be negative. Therefore, x cannot equal -5 under the given conditions.

Conclusion

The value -5 cannot be a solution for x because it leads to an invalid condition regarding the absolute value of y. Other options, such as 2 and -2, allow for valid solutions that adhere to the equation while respecting the constraint x ≠ y. Thus, -5 is the only option that definitively cannot be equal to x.

The probability that a household subscribing to cable television also subscribes to home phone service is 14/18.

In Malia's data, 14 out of the 18 households that subscribe to cable television also subscribe to home phone service. Therefore, the probability is calculated as the ratio of these two numbers.

A) 14/18

This option is correct because it accurately represents the fraction of households that subscribe to both cable television and home phone service out of those that subscribe to cable television. The count of households subscribing to both services is 14, and the total number of households with cable is 18.

B) 14/26

This option is incorrect as it suggests that there are 26 households that subscribe to cable television. The total number who subscribe to cable television is only 18, making this ratio inaccurate.

C) 18/36

This option is also incorrect. While 18 is the number of households that subscribe to cable television, this fraction does not reflect the number of households that also subscribe to home phone service. The total number of households is 36, but the relevant total for the probability calculation must be only those who subscribe to cable.

D) 14/36

This option is incorrect because it implies that the probability is based on the total number of all households (36) rather than just those that subscribe to cable television (18). The numerator should reflect the 14 households that subscribe to both services.

Conclusion

The correct answer, 14/18, effectively captures the probability of a household subscribing to home phone service given that they already subscribe to cable television. All other options fail to provide the correct context or numbers necessary for calculating the probability as required by the question.

The area of the circle is increased by 800% when the radius is tripled.

When the radius of a circle is tripled, the area increases by 800%. This is because the area of a circle is proportional to the square of the radius, and tripling the radius results in an area that is nine times greater than the original area, which translates to an 800% increase.

A) 200%

Option A is incorrect because a 200% increase would only imply that the area becomes three times the original area, not accounting for the squared relationship between the radius and the area.

B) 300%

Option B is also incorrect. A 300% increase would mean the area is four times the original area, which is not the case when the radius is tripled, as the area actually becomes nine times the original.

C) 400%

Option C is incorrect as well, as a 400% increase would indicate that the area is five times the original area. This does not align with the mathematical relationship governing the area of a circle when the radius is increased.

D) 800%

Option D is correct. Tripling the radius results in the area becoming nine times larger, which equates to an increase of 800% over the original area, confirming the mathematical principle that area scales with the square of the radius.

Conclusion

In conclusion, the correct answer is 800% because tripling the radius of a circle leads to an area that is nine times the original, resulting in an increase of 800%. The other options fail to accurately represent the geometric relationship between radius and area, which is fundamental to understanding this problem.

|7 - 5| is equal to |5 - 7|.

The expression |7 - 5| simplifies to 2, and the expression |5 - 7| also simplifies to 2, making them equal.

A) |7| + |-5|

This option evaluates to |7| + |-5|, which is 7 + 5 = 12. This is incorrect as it does not equal 2, the value of |7 - 5|.

B) |5| - |7|

Calculating this gives |5| - |7|, which simplifies to 5 - 7 = -2. The absolute value of -2 is 2, but this does not represent the same expression as |7 - 5| and is not equal to it.

C) |5 - 7|

This option evaluates to |5 - 7|, which simplifies to | -2 |, yielding a value of 2. This is correct as it matches the value of |7 - 5|.

D) |-7 + (-5)|

This option simplifies to |-7 - 5|, resulting in |-12|, which equals 12. This is incorrect since it does not equal 2.

Conclusion

The correct answer is C) |5 - 7|, as both |7 - 5| and |5 - 7| evaluate to the same numerical value of 2. The other options either yield different values or do not represent the same calculation, confirming that C is the only correct choice.

The price of each shirt is 75 - n/2 dollars.

To find the price of each shirt in terms of n, we start by setting up the equation based on John's purchases. He bought 2 shirts and 1 pair of pants for a total of 75 dollars, where the pants cost n dollars. This leads us to the equation 2s + n = 75, allowing us to solve for the price of each shirt (s) as 75 - n/2.

A) 75/2

This option suggests that each shirt costs 75/2 dollars, which would imply that the total for two shirts is 75 dollars. However, this does not account for the cost of the pants, leading to an incorrect total that does not match the given information.

B) 75 + n/2

This option indicates that each shirt costs 75 + n/2 dollars. If this were the case, the total cost for two shirts would exceed the total of 75 dollars when the cost of the pants is included, making this option impossible within the constraints of the problem.

C) 75 - n/3

This choice suggests that each shirt is priced at 75 - n/3 dollars. However, substituting this into the equation for total cost would not yield a valid solution, as it would lead to a total that does not satisfy the original equation of 2s + n = 75.

D) 75 - n/2

This option accurately represents the relationship established by the equation derived from John's purchases. By substituting the expression for s back into the equation, we confirm that the total cost of the two shirts and the pants indeed totals 75 dollars, validating this choice as correct.

Conclusion

The correct answer, 75 - n/2, provides a clear and precise calculation of the price of each shirt based on the total expenditure of 75 dollars and the cost of the pants. All other options misinterpret the relationship between the cost of the shirts and pants, leading to incorrect totals that do not align with the scenario presented.

The area of the rectangle is decreased by 9%.

When the length of a rectangle is increased by 30% and the width is decreased by 30%, the overall effect on the area results in a decrease of 9%.

A) It is increased by 60%.

This option is incorrect because an increase of 30% in length does not lead to a 60% increase in area when coupled with a 30% decrease in width. The calculations show that the area actually decreases.

B) It is unchanged.

This option is also incorrect. A change in dimensions of the rectangle (increasing length and decreasing width) results in a net change in area. Therefore, the area cannot remain unchanged.

C) It is decreased by 15%.

This choice is incorrect because the calculations show that the area decreases by only 9%. A 15% decrease does not accurately reflect the combined effects of the dimensional changes.

D) It is decreased by 9%.

This option is correct. The increase in length by 30% and the decrease in width by 30% lead to a reduction in area, which can be calculated to determine that the net effect is a 9% decrease.

Conclusion

The correct answer, a 9% decrease in area, results from the interplay of the increased length and decreased width. Other options fail to accurately reflect the mathematical relationship between the changes in dimensions and their impact on the area, confirming that D is the only valid conclusion based on the given conditions.

The price of each shirt in terms of n is (75-n)/2.

To find the price of each shirt, we first express the total price of the two shirts and the pants. The total cost is 75 dollars, where the pants cost n dollars. Thus, the cost of the two shirts combined is 75 - n dollars, leading to the price of each shirt being (75 - n)/2.

A) (75-n)/2

This option correctly calculates the price of each shirt by considering the total price of the two shirts as 75 - n. Dividing by 2 gives the price per shirt, making this option accurate.

B) (75+n)/2

This option incorrectly adds n to 75 before dividing by 2, which does not reflect the total cost of the shirts. The addition of n does not correspond to the price structure described in the question, rendering this option incorrect.

C) (75-n)/3

This option incorrectly divides the total cost of the two shirts (75 - n) by 3 instead of 2. Since there are only two shirts, this miscalculation leads to an inaccurate price per shirt, making this option incorrect.

D) (75-n)/2

This option is the same as A, correctly determining the price of each shirt as (75 - n)/2. It accurately reflects the total cost of the shirts based on the information provided in the question, confirming its correctness.

Conclusion

The correct answer is D, which accurately represents the price of each shirt in relation to the cost of the pants. Options A and D are identical and correct, while B and C fail to adhere to the problem's requirements by miscalculating the distribution of costs. Thus, D is definitively the valid answer based on the given conditions.

There could be none or more than one solution to the system of equations.

The system of equations can either be inconsistent, having no solutions, or dependent, leading to infinitely many solutions. Since the second equation is a multiple of the first, it suggests that the lines represented by these equations could either coincide or be parallel, impacting the number of solutions.

A) I only

This option states that there are none of the solutions to the system. However, since the equations can either be consistent or inconsistent, it is incorrect to claim that there are only no solutions without considering the possibility of infinitely many solutions.

B) III Only

This choice indicates that there are more than one solution. While it is true that if the equations are dependent there would be infinitely many solutions, this option fails to account for the possibility of having no solutions at all, which could occur if the lines are parallel.

C) I or II

This option suggests that there could be either none or one solution. However, it overlooks the possibility of having infinitely many solutions, which is a valid scenario when the equations are dependent. Thus, this choice does not encompass all potential outcomes.

D) I or III

This choice accurately captures the scenarios that can arise from the equations. It recognizes that the system can be inconsistent, resulting in no solutions, or dependent, leading to more than one solution. This comprehensive view makes it the correct answer.

Conclusion

The correct answer, D, effectively encompasses the various possibilities of the system of equations, acknowledging both the potential for no solutions and infinitely many solutions. Options A, B, and C each fail to represent all possible outcomes, making D the only option that accurately reflects the nature of the given equations.

2(3^(3/2))

The expression 3^(5/2) - 3^(3/2) can be simplified by factoring out the common term, resulting in 2(3^(3/2)).

A) 1

This option is incorrect because the expression 3^(5/2) - 3^(3/2) simplifies to a value greater than 1. The calculation of the terms involved shows that their difference does not equal 1.

B) 2

Option B is also incorrect as the simplified expression does not evaluate to 2. Instead, the correct simplification leads to a multiple of 3^(3/2), which is significantly larger than 2.

C) 3

This choice is incorrect as well since the difference between 3^(5/2) and 3^(3/2) yields a value that is not equal to 3. The expression simplifies to a value that is dependent on 3^(3/2).

D) 3^2

Option D is incorrect because 3^2 equals 9, which is far greater than the result of the simplification of the original expression. The calculation does not support this value as a result.

E) 2(3^(3/2))

This option is correct as it accurately reflects the simplified form of the expression 3^(5/2) - 3^(3/2). By factoring out 3^(3/2), the expression simplifies to 2 times 3^(3/2), making this the correct answer.

Conclusion

The correct answer, 2(3^(3/2)), is derived from the proper simplification of the given expression. All other options fail to represent the mathematical operations involved, leading to incorrect values that do not align with the result of the subtraction. Thus, option E stands out as the definitive correct choice.

st=215

The equation st=215 effectively represents the relationship between speed, time, and distance, where s is the average speed in miles per hour, t is the time in hours, and the product of speed and time equals the distance traveled, which is 215 miles.

A) st=215

This option correctly formulates the relationship between speed, time, and distance based on the formula Distance = Speed × Time. Here, st accurately represents the distance of 215 miles traveled by Susan, making it the right choice.

B) 215+t=s

This option is incorrect because it wrongly rearranges the relationship. It suggests that speed (s) equals the distance (215) plus time (t), which does not conform to the basic distance formula.

C) s/t=215

This option is also incorrect as it represents the average speed divided by time, yielding a different value rather than the relationship between distance, speed, and time. The correct formula involves multiplication, not division.

D) s+t=215

This option misinterprets the relationship by suggesting that speed plus time equals distance, which is not mathematically accurate. The sum of speed and time does not equate to distance traveled.

Conclusion

The equation st=215 is the only option that accurately reflects the fundamental relationship of distance traveled through the multiplication of speed and time. All other options fail to represent this relationship correctly, either through incorrect operations or misinterpretations of the variables involved.