SOLUTION: {{{sqrt(b-5) - sqrt(b+7) <= 4}}}

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Question 202636This question is from textbook Algebra 2
: sqrt%28b-5%29++-++sqrt%28b%2B7%29++%3C=++4 This question is from textbook Algebra 2

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The simplest way to solve this is by using logic (as opposed to using Algebra):
  1. Which square root, sqrt%28b-5%29 or sqrt%28b%2B7%29 is larger, regardless of the value of "b"? It should be obvious that the second one is always going to be larger.
  2. When you subtract a larger number from a smaller one, like in the left side of your inequality, what do you always get? Answer: a negative number.
  3. At this point we have determined that the left side of the inequality will always be a negative number. And which negative numbers are less then or equal to 4? Answer: All of them!
  4. Since the left side must be a negative number and since all negative numbers are less than or equal to 4, we might be thinking that the solution to your problem is "All real numbers". But there are implied limits to the values "b" can have. We must avoid negative numbers inside the square roots. So we must ensure that both (b-5) and (b+7) are zero or positive.
  5. When you need two numbers to be zero or positive, make sure the smaller one is zero or positive. If the smaller one is zero or positive, the larger one would have to be positive. (Think about this if it isn't obvious.)Since (b-5) is always less than (b+7), all we need to do, to ensure that:
    %28b-5%29+%3E=+0
    Adding 5 to both sides we get
    b+%3E=+5

So our solution is b+%3E=+5.