If a >b and a^2<b^2 then which of the following must be always true?
A. a<0
b. 0<a<1
C. b<0
D none of these
Do a test... that is choose a>b
6>5
but then a^2 would not be less than b^2
So choose a= a negative number... say -1
and b = -2
Does that help ??
So what is the answer
To determine which option must always be true, let's examine the given conditions:
1. a > b: This means that a is greater than b.
2. a^2 < b^2: This implies that the square of a is less than the square of b. By taking the square root of both sides of the inequality, we find that a < b.
Now, let's analyze the options:
A. a < 0: This option states that a is less than 0. However, we don't have enough information to conclude that a is always negative. For example, if a = 2 and b = 1, then a > b but a is not less than 0. Therefore, option A is not necessarily true.
B. 0 < a < 1: This option states that a is between 0 and 1. Again, we don't have sufficient information to determine that a will always fall within this range. For example, if a = 10 and b = 1, then a > b but a is not between 0 and 1. So, option B is not necessarily true.
C. b < 0: This option states that b is less than 0. Similarly, we can't conclude that b will always be negative based on the given conditions. For example, if a = 1 and b = -2, then a > b but b is not less than 0. Therefore, option C is not necessarily true.
D. None of these: Based on our analysis, we can conclude that none of the options always satisfy the given conditions. So, option D is correct.
Therefore, the answer is option D: none of these.