If -4 < x < 7 and -6 < y < 3, which of the following specifies all the

I think the answer should be -42<xy<24, which is not one of the options.

To find the range of xy, we need to find the minimum this number can be. Since from the ranges given for both x and y cover both positive and negative numbers, the range of xy would also range from negative number to positive number.

To find the minimum, we have to find the maximum negative number possible => one of the x and y is positive and the other is negative and the magnitude of the product is maximum. This can be achieved when x~7 (x cannot be equal to 7 but can be as close as possible) and y~-6. This combination gives xy~-42

To find the maximum, we have to find a positive number =>either both x and y are negative OR both x and y are positive => we compare the products (-4*-6) and (7*3) =>maximum of these is 24.

IF -4 < X < 7 and -6 < y < 3, which of the following specifies all the possible values of xy?

A. -42 < XY <21
B. -42 < XY < 24
C. -28 < XY < 18
D. -24 < XY < 21
E. -24 < XY < 24

OA

So I got this wrong on the exam because I was pressed for time and was multiplying the 2 extreme positive and negative numbers and not finding my answer. Also I since it say x and y were LESS than, I looked at XY with the idea that x was -3 to 6 inclusive and Y was -5 to 2 inclusive.

Anyway, when I reviewed, I drew out the number line on each and could clearly see the right answer. My question is "is there a faster way to do this than diagramming it all out or is that generally the best/fastest approach?

Multiply all the possible combinations in such cases:

-4 * 7 = -28

-4 * -6 = 24 >>> Highest

-4 * 3 = -12

7 * -6 = -42 >>> Lowest

7 * 3 = 21

Answer = B

Conventional method:

Draw the number line as below:

Hi

This is min/max question. To find the range of xy, you need to find MAX xy and MIN xy

The easiest way is to test and see what max and min are.
Max = -4*-6 = 24
Min = -6*7 = -42

Range: -42 < xy < 24

Hope it helps.

This is a question that tests your observation of the following facts:

- x - = +
- x + = -
+ x - = -
+ x + = +

Thus, you need to now multiply the extreme values of x (-4,7) with extreme values of y (-6, 3) to get :

-4 x -6 = 24
-4 x 3= -12
7 x - 6= -42
7 x 3 = 21

Thus you see the minimum value = -42 and the maximum value = 24 . All other combinations of xy will give you values within these 2 values.

Hence the range of xy: -42 < xy < 24

B is the correct answer.

Hope this helps.

Minimum value of xy is the product in absolute terms of x and y that give a min value = 7 * (-6) = - 42
Maximum value of xy is the product in absolute terms of x and y that give a max value = (-4) * (-6) = 24
thus the range in which all possible values of xy can lie is -42< xy <24
correct option - B

If -4 < x < 7 and -6 < y < 3, which of the following specifies all the possible values of xy?

Initially consider all posssible values of xy, we can decide the ranges later
xy=24, -12, -42, 21
now pick the extremes
-42<xy<21

-4<x<7 : -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7 : with boundary values
-6<y<3 : - 6, -5, -4, -3, -2, -1, 0, 1, 2, 3 : with boundary values

what is the minimum and maximum product valud of XY we can get here
Start multiplying the boundary values :
-4*-6 = 24
7*3 = 21
-7 * -6 = -42
-4 * 3 = -12

out of these values take the minimum and maximum values ; -42< xy < 24

Minimum value of xy is -6*7=-42
Maximum value of xy is -6*-4=24
answer is B

To determine the largest possible value of xy, we either multiply together the two smallest negative values or the two largest positive values. Since (-4)(-6) = 24 and (7)(3) = 21, and 24 > 21, we see that the largest possible product of x and y is less than 24.

To determine the smallest value of xy, we multiply the largest positive number by the smallest negative number. Thus, the product of x and y must be greater than (7)(-6) = -42. Thus:

-42 < xy < 24

Answer: B

possible values of x = -3, -2, -1, 0 ,1, 2, 3, 4, 5, 6
Possible values of y = -5 ,-4 ,-3 ,-2, -1, 0, 1, 2

values of xy ranges between -30 to 15

With this logic, I chose option A.

My question here is, why are we assuming xy could be 24 (-6).(-4) because it doesn't say inclusive? Why are we considering -6 and -4 as possible values of y and x respectively?

The fastest and most accurate method is often just to list the possibilities for xy.

Multiply each value of x's range (low end, high end), by each value of y's range (low and high).

-4 < x < 7
-6 < y < 3

(x)(y)?

(-4)(-6) = 24
(-4)(3) = -12
(7)(-6) = -42
(7)(3) = 21

The smallest is -42
The greatest is 24

-42 < xy < 24

Answer B

Whatever the case, to maximize xy (to find the greatest number, the upper end of the inequality for xy):
-- find one end value of x which, when multiplied by one end value of y, is the greatest number. Do not assume that rightmost values for x and y will produce the greatest product. That is trap Answer A here. 24 > 21

In this case, for example: the greatest product of (x*y) consists of multiplying x's and y's SMALLEST numbers (i.e., the numbers -4 and -6, which mark the low end of their respective ranges).

To minimize, to find the smallest number xy can be:
-- use one of the end numbers from x's range which, when multiplied by an end number from y's range, yields the smallest number
-- in this case, the smallest number is -42, where -42 is the "most negative" number, the one farthest to the left of zero in the number line.

Always check for the 4 boundary conditions.
See the procedure in the Sketch.

Let's examine the EXTREME VALUES Of x and y and see what happens.

If we want to MINIMIZE the value of xy, we need to examine what happens when 1 EXTREME value is positive and 1 EXTREME value is negative.
case a: x = -4 and y = 3, in which case xy = -12
case b: x = 7 and y = -6, in which case xy = -42
Great, so xy is MINIMIZED when x = 7 and y = -6
Of course, we're told that x < 7 and y > -6, but that's fine. Basically, this means that xy > -42

At this point, we know that the correct answer must be either A or B.

Next, if we want to MAXIMIZE the value of xy, we need to examine what happens when both EXTREME values are positive or both are negative.
case c: x = -4 and y = -6, in which case xy = 24
case d: x = 7 and y = 3, in which case xy = 21
Great, so xy is MAXIMIZED when x = -4 and y = -6
Of course, we're told that x > -4 and y > -6, but that's fine. Basically, this means that xy < 24

So, as you can see, -42 < xy < 24

Answer: B

Cheers,
Brent

The multiplications of xy and will be
24,21,-42,-12
So, the range will be of all values
-42 < xy < 24

The Answer is B.

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