Question 180067This question is from textbook elementry and intermediate algebra
: 1. Explain the following in your own words: (a + b)2 ≠ a2 + b2. Give a numerical example to illustrate this.
3. Add -5 to both sides of the inequality 7 > 1. Then divide both sides of the inequality -25 > -30 by -6. Explain in your own words what happens in both examples and when the direction of the inequality symbol is reversed.
This question is from textbook elementry and intermediate algebra
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
I can explain these things in my words, but if you are going to use this for your homework, you'll have to change them and still manage to communicate the same ideas in order for you to properly answer the questions.
^2 \neq a^2 + b^2)
Obviously, these two expressions are equal if either a, b, or both are 0. The question is, do there exist any non-zero values for a and b that exhibit the property ^2 = a^2 + b^2) ?
Let's assume that there exists a number  and a number  such that .
We can add  to both sides:
^2 - b^2 = a^2) .
Then we can factor the difference of two squares on the left:
 - b)((a + b) + b) = a^2) .
Collect terms and distribute:
 = a^2) .
 .
Now add  to both sides:
 .
According to the Zero Product Rule, if  then either  or  , but that contradicts our original assumption that and . Hence, reductio ad absurdum there are no non-zero values for a and b such that . Therefore, ^2 \neq a^2 + b^2 \text{ } \math \forall a \neq 0 \text{ and } \math b \neq 0)
Having said all of that, finding a numerical example to illustrate should be a fairly simple task. I'll leave that part to you.
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Add -5 to the inequality 7 > 1
 which is a true statement without changing the sense of the inequality.
Then divide both sides of the inequality -25 > -30 by -6.
Which is now a false statement. Since we divided by a number less than zero, the sense of the inequality must be reversed. This is also true when you multiply by a negative, since dividing by a number is the same as multiplying by the reciprocal.
Having reversed the sense of the inequality, the statement is again true.
John
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