InequalitiesDate: 08/01/97 at 02:44:30 From: Eytan Reiss Subject: Greater than How do I deal with an equation with a greater than > or less than < sign? I tried treating it like an equals sign but got a ridiculous answer. ex: 7+(X=7) > 3 Thanks and sincerely yours, Eytan Reiss Date: 08/07/97 at 14:13:51 From: Doctor Rob Subject: Re: Greater than Mostly you can work with inequalities in the same way you do with equalities, but there are some important differences. The rules are more or less like this: Let a, b, x, y, and z be any real numbers. If x > y and y > z, then x > z. If x > y and a = b, then x + a > y + b. If x > y and a = b, then x - a > y - b. If x > y and a = b > 0, then x*a > y*b. If x > y and a = b > 0, then x/a > y/b. If x > y and a = b < 0, then x*a < y*b. If x > y and a = b < 0, then x/a < y/b. If x > y and a > b, then x + a > y + b. If x > y and a < b, then nothing can be said about x + a and y + b. If x > y and a < b, then x - a > y - b. If x > y and a > b, then nothing can be said about x - a and y - b. If x > y and a > b > 0, then x*a > y*b. If x > y and 0 > a > b, then nothing can be said about x*a and y*b. If x > y and a < b < 0, then x*a < y*b. If x > y and 0 < a < b, then nothing can be said about x*a and y*b. If x > y and a > b > 0, then nothing can be said about x/a and y/b. If x > y and 0 > a > b, then x/a < y/b. If x > y and a < b < 0, then nothing can be said about x/a and y/b. If x > y and 0 < a < b, then x/a > y/b. If x^2 < a > 0, then -Sqrt[a] < x < Sqrt[a]. If x^2 > a > 0, then either x < -Sqrt[a] or else x > Sqrt[a]. All the above are still true if you replace > with >= or replace < with <=. All the statements made above are true essentially because of the laws of signs for addition, subtraction, multiplication, and division. For example, x > y means x = y + p for some positive p (i.e. p > 0). If a = b > 0, then a and b are positive. Then a*x = b*(y + p) = b*y + b*p. Since p is positive and b is positive, p*b is positive (law of signs for multiplication), so a*x > b*y (since to get from b*y to a*x you add something positive). The rest are similar. I hope this helps. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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