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If x and y are integers such that (x+1)^2 is less than or equal to 36 and (y-1)^2 is less than 64, what is the sum of the maximum possible value of xy and the minimum possible value of xy?
(A) -16
(B) -14
(C) 0
(D) 14
(E) 16
_________________
(A) -16
(B) -14
(C) 0
(D) 14
(E) 16
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Posts: 9
Since X and Y are integers and we are trying to min/max XY we need to look for the extremes of the equations (either negative or positive)
(x+1)^2<= 36, therefore (x+1) has to be 6 or -6 and X=5 and X=-7
(y-1)^2 <64, therefore (y+1) has to be less than 8 or greater than -8. As such, either Y=8 or Y=-6
The maximum possible value of xy is X=-7 and Y=-6 (42) and the minimum possible value of xy is X=-7 and Y=8 (-56).
Therefore, 42+ (-56)= -14
(x+1)^2<= 36, therefore (x+1) has to be 6 or -6 and X=5 and X=-7
(y-1)^2 <64, therefore (y+1) has to be less than 8 or greater than -8. As such, either Y=8 or Y=-6
The maximum possible value of xy is X=-7 and Y=-6 (42) and the minimum possible value of xy is X=-7 and Y=8 (-56).
Therefore, 42+ (-56)= -14
General Discussion
(x+1)^2 <= 36
x <= 5
x >= -7
(y-1)^2 < 64
y < 9
y > -7
Max possible value of xy is -7 × -6 = 42
minimum possible value of xy is -7 × 8 = -56
-56 + 42 = -14
Answer : B
x <= 5
x >= -7
(y-1)^2 < 64
y < 9
y > -7
Max possible value of xy is -7 × -6 = 42
minimum possible value of xy is -7 × 8 = -56
-56 + 42 = -14
Answer : B
Bunuel
(x+1)^2<=36
-6<=x+1<=6
-7<=x<=5
(y-1)^2<=64
-8<=y-1<=8
-7<=y<=9
Max xy=-7*-7=49
Min xy=-7*9=-63
49-63=-14
Answer: B
SherLocked2018
Joined: 25 Nov 2014
Last visit: 03 May 2017
Posts: 92
Concentration: Entrepreneurship, Technology
GMAT 1: 680 Q47 V38
GPA: 4
(x+1)^2 <= 36 gives, -6<=x+1<=6 gives, -7<=x<=5
(y-1)^2 <= 64 gives, -7<=y-1<=7 gives, -6<=y<=8
max value of xy = 42 , and min value of xy = -56
Thus sum = -14
Ans B.
(y-1)^2 <= 64 gives, -7<=y-1<=7 gives, -6<=y<=8
max value of xy = 42 , and min value of xy = -56
Thus sum = -14
Ans B.
Given
\((x+1)^2 \leq(36)\)
\(-6 \leq x+1 \leq 6\)
\(-7 \leq x \leq 5\)
\((y-1)^2 \leq(64)\)
\(-8 \leq y-1 \leq 8\)
\(-7 \leq y \leq 9\)
Max value of \(xy = -7*-7\)
Min value of \(xy = -7*9\)
Sum \(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= -7 (-7+9) = -14\)
Answer B
\((x+1)^2 \leq(36)\)
\(-6 \leq x+1 \leq 6\)
\(-7 \leq x \leq 5\)
\((y-1)^2 \leq(64)\)
\(-8 \leq y-1 \leq 8\)
\(-7 \leq y \leq 9\)
Max value of \(xy = -7*-7\)
Min value of \(xy = -7*9\)
Sum \(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= -7 (-7+9) = -14\)
Answer B
Bunuel
VERITAS PREP OFFICIAL SOLUTION
To get the sum of the maximum and minimum possible values of xy, we need to know the maximum and minimum values of xy. For those, we need to find the values that x and y can take. So first, we should review the information given:
(x + 1)^2 <= 36
(y – 1)^2 < 64
We need to find the values that x and y can take. There are many ways of doing that. We can solve the inequality using the wave method discussed in this post or using the concept of absolute values. Let’s discuss both the methods.
Wave method to solve inequalities:
Solve for x: (x + 1)^2 <= 36
(x + 1)^2 – 6^2 <= 0
(x + 1 + 6)(x + 1 – 6) <= 0
(x + 7)(x – 5) <= 0
-7 <= x <= 5 (Using the wave method)
Solve for y: (y – 1)^2 < 64
(y – 1)^2 – 8^2 < 0
(y – 1 + 8)(y – 1 – 8) < 0
(y + 7)(y – 9) < 0
-7 < y < 9 (Using the wave method)
Or you can solve taking the square root on both sides
Solve for x: (x + 1)^2 <= 36
|x + 1| <= 6
-6 <= x + 1 <= 6 (discussed in your Veritas Algebra book)
-7 <= x <= 5
So x can take values: -7, -6, -5, -4, … 3, 4, 5
Solve for y: (y – 1)^2 < 64
|y – 1| < 8
-8 < y – 1 < 8 (discussed in your Veritas Algebra book)
-7 < y < 9
So y can take values: -6, -5, -4, -3, … 6, 7, 8.
Now that we have the values of x and y, we should try to find the minimum and maximum values of xy.
Note that the values of xy can be positive as well as negative. The minimum value will be the negative value with largest absolute value (largest negative) and the maximum value will be the positive value with the largest absolute value.
Minimum value – For the value to be negative, one and only one of x and y should be negative. Focus on the extreme values: if x is -7 and y is 8, we get xy = -56. This is the negative value with largest absolute value.
Maximum value – For the value to be positive, both x and y should have the same signs. If x = -7 and y = -6, we get xy = 42. This is the largest positive value.
The sum of the maximum value of xy and minimum value of xy is -56 + 42 = -14
Answer (B)
Try to think of it in terms of a number line. x lies in the range -7 to 5 and y lies in the range -6 to 8. The range is linear so the end points give us the maximum/minimum values. Think of what happens when you plot a quadratic – the minimum/maximum could lie anywhere.
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Bunuel
\((x+1)^2\leq{36}\);
\({-\sqrt{36}}\leq{x+1}\leq{\sqrt{36}}\);
\({-6}\leq{x+1}\leq{6}\);
\({-7}\leq{x}\leq{5}\).
\((y-1)^2<{64}\);
\({-\sqrt{64}}<{y-1}<{\sqrt{64}}\);
\({-8}<{y-1}<{8}\);
\({-7}<{y}<{9}\), as \(y\) is an integer we can rewrite this inequality as \({-6}\leq{y}\leq{8}\).
We should try extreme values of \(x\) and \(y\) to obtain min and max values of \(xy\):
Min possible value of \(xy\) is for \(x=-7\) and \(y=8\) --> \(xy=-56\);
Max possible value of \(xy\) is for \(x=-7\) and \(y=-6\) --> \(xy=42\).
The sum = -56 + 42 = -14.
Answer: B.
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Since it has been given that : \(\left(x+1\right)^2\ =\ 36\)
The possible values of x are : x = 5, x = -7.
Since \(\left(y-1\right)^2\ =\ 64\)
Similarly, the possible values of y are 9, -7.
The maximum value of xy is when : x = -7, y = -7. The maximum value of x*y = 49
The minimum value of xy is when : x = -7, y = 9. The minimum value of x*y = -63.
The sum is 49 - 63 = -14.
The possible values of x are : x = 5, x = -7.
Since \(\left(y-1\right)^2\ =\ 64\)
Similarly, the possible values of y are 9, -7.
The maximum value of xy is when : x = -7, y = -7. The maximum value of x*y = 49
The minimum value of xy is when : x = -7, y = 9. The minimum value of x*y = -63.
The sum is 49 - 63 = -14.
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