simple inequality problem: integer solutions for 4=<a<7, -5<b=<3

ketanco

New member
Joined
Oct 15, 2018
Messages
26
been a while since i looked at this subject so need help

4=<a<7
-5<b=<3

how many integer values can a.b take?

answer should be 55

but when i multiply -4 and 5 lower limit is 20 and higher limit is 3*7=21 so it makes 19+20=41.. how can it be 55? missing something here.... 41 wasnt even given in the answer choices

(didnt know how to write "greater than or equal to" sign, but i am sure you get the point)
 
been a while since i looked at this subject so need help
4=<a<7
-5<b=<3
how many integer values can a.b take?
answer should be 55
WHY is this true \(\displaystyle \bf{-35<a\cdot b<21}~?\)
 
been a while since i looked at this subject so need help

4=<a<7
-5<b=<3

how many integer values can a.b take?

answer should be 55

but when i multiply -4 and 5 lower limit is 20 and higher limit is 3*7=21 so it makes 19+20=41.. how can it be 55? missing something here.... 41 wasnt even given in the answer choices

(didnt know how to write "greater than or equal to" sign, but i am sure you get the point)

You can't just multiply two inequalities together, when negative numbers are possible. The fact that a >= 4 and b > -5 does not imply that ab > -20; for example, if a = 5 and b = -5, you get ab = -25, which is not greater than 20. Think about what happens when you multiply an inequality by a negative number.

The lower limit turns out to be -5*7, not -5*4. Can you see why?
 
You can't just multiply two inequalities together, when negative numbers are possible. The fact that a >= 4 and b > -5 does not imply that ab > -20; for example, if a = 5 and b = -5, you get ab = -25, which is not greater than 20. Think about what happens when you multiply an inequality by a negative number.

The lower limit turns out to be -5*7, not -5*4. Can you see why?

Yes i see... so when there are negatives we must test every possibility
 
Yes i see... so when there are negatives we must test every possibility

To be very careful, I might take it like this:

There are two cases, depending on whether b is positive or negative. (I'll include zero with positive.)

First, we have 4 <= a < 7 and 0 <= b <= 3. In this case, 0 <= ab < 21.

Second,
we have 4 <= a < 7 and -5 < b < 0. In this case, -35 < ab < 0.

Putting it together, -35 < ab < 21. Any integer in this interval is possible. The total number of them is 21 - (-35) - 1 = 55.
 
Top