For all real numbers x and y, if x is not equal to y, x>0,y>0 then (x/y)+(y/x)>2
For all real numbers x and y, if x is not equal to y, x>0,y>0 then...
Prove that x^2+xy+y^2≥0 for all real numbers, x and y. Find the values that result in equality.
Suppose R is the relation defined on all real numbers by for all real numbers x,y (xRy if |x-yl3) Then for real numbers x and y, xR2y iff
Show that [x + y] ≥ [x] + [y] for all real numbers x and y. Show by induction that if a and b are integers such that a | b, then a^k | b^k for every positive integer k.
If x and y are real numbers and if xy=0, then either x=0 or y=0 Give two examples of a typical problem or calculation in high school algebra in which this role of 0 is used
x+3 2x Define f(x) for all real numbers x = 0. Is f a one-to-one function? Prove or give a counterexample. (Note that the write-up of the proof or counterexample should only have a few of sentences.) If the co-domain is all real numbers not equal to 1, is f an onto function? Why or why not? (Note this problem does not require a full proof or formal counterexample, just an explanation.)
Show that if x and y are real numbers, x2 + y2 >= 2xy and (x + y)2 >= 4xy; When does equality hold (with proof)? Show that if x and y are real numbers, x2 + y2 2xy and (x y) 2Hry. When does equality hold (with proof)?
9. Let x,y > 0 be real numbers and q, r E Q. Prove the following: (а) 29 > 0. 2"а" and (29)" (b) x7+r (с) г а — 1/29. 0, then x> y if and only if r4 > y (d) If q (e) For 1, r4 > x" if and only if q > r. For x < 1, x4 > x* if and only if q < r.
A. Let x and y be two real numbers such that y - 2x = 20 and (3 - x)(y + 2) is a maximum. Find x and y. B. Suppose that one of these numbers is 6 what is the second one? Why?
Prove that for any two real numbers x and y, |x + y| ≤ |x| + |y|. Hint: Use the previously proven facts that for any real number a, |a|≥ a and |a|≥−a. You should need only two cases.
[9] Given any two real numbers x and y such that x < y, show that there exists a rational number q such that x < a <y.