<li>If -5 < x < 8, and -10 < y < 4, which of the following describes the range of all possible values of xy?</li>
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<p>(A) -20 < xy < 32
(B) -50 < xy < 32
(C) -50 < xy < 80
(D) -80 < xy < 20
(E) -80 < xy < 50</p>
<p>[Please explain this very carefully.]</p>
<p>take the bounds of x and y and multiply.</p>
<p>8<em>-10 gives the smallest possible number.
-5</em>-10 gives the largest possible number.</p>
<p>-80<xy<50 E</p>
<p>I tried doing that. But I did -5<em>-10 and 8</em>4. Why do must you do 8<em>-10 and -5</em>-10? I didnt get that.</p>
<p>You need to remember to check all combinations of the bounds. Since a negative times a negative is a positive, the upper bound of xy may occur (and this time it actually does occur) when you multiply the lower bound of x and the lower bound of y. Just multiply out all four combinations (50, -20, -80, 32) and see what the max and min are</p>
<p>If we consider all possible values of x and y as the coordinates of some point P(x,y), we can look at xy as the area of rectangle with two of its vertices in the origin and point P (negative or 0 areas should not scare us).</p>
<p>In the first quadrant the area xy can vary from 0 at p(0,0) all the way up to
4<em>8 =32 without reaching 32 at P(8,4).
In the second quadrant the area xy varies from 0 to "almost" (-5)</em>4 = -20.
In the third quadrant (-5)<em>(-10)=50, so 0<=xy<50.
And finally, in the fourth quadrant, since 8</em>(-10)=-80, -80<xy<=0.</p>
<p>Summarizing, -20 is the lower bound and 80 is the upper bound for xy, or
-20<xy<80.</p>
<p>Very similar question:
0 <= x <=8,
-1<= y <= 3
xy?</p>
<p>It's actually even easier, since there only two quadrants to look into (I and IV), and lower and upper bounds are included.</p>
<p>In the first quadrant point P(x,y) can be anywhere inside or on the sides of rectangular with the vertices (0,0), (0,3), (8,3), and (8,0) ,
so the area xy of rectangle with two of its vertices in the origin and point P(x,y) is between 0 for P(0,0) and 8*3 = 24 for P(8,3), including 0 and 24, or
0<= xy <= 24.</p>
<p>In the second quadrant 8*(-1) = -8, so -8<= xy <=0,</p>
<p>Conclusion: all possible values of xy are -8<= xy <= 24.
Answer E.</p>
<p>Now that we know how it works, in questions like this one you can just select an answer with the widest range for xy. Make sure though that you can get its bounds by multiplying some of the x and y bounds.
Otherwise, if -24<= xy <= 24 were one of the answer choices for our question, you'd choose it - and loose it (1 1/4).</p>