SOLUTION: -What does it mean for two algebraic expressions to be equivalent? -What does it mean for a set of numbers to be a solution for an equation or inequality?

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Question 151: -What does it mean for two algebraic expressions to be equivalent?
-What does it mean for a set of numbers to be a solution for an equation or inequality?
Answer by KMST(5484) About Me  (Show Source):
You can put this solution on YOUR website!
Two algebraic expressions are equivalent if they have the same value when the same value is used for each variable.
For example,

%28x%2B1%29%28x-1%29 is equivalent to x%5E2-1 because
for every value of x both expressions result in the same number.

%28x%2By%29%28x-y%29 and x%5E2-y%5E2 are equivalent
because for each and every pair of %22%28+x+%2C+y+%29%22 values,
the value for both expressions is the same.

%28x%2B1%29%28x-1%29%2F%28x-1%29 is not equivalent to x%2B1
because for x=1 the first expression does not exist, while x%2B1=1%2B1=2

A set of numbers is the solution set for an equation or inequality if and only if
those are the only numbers make the equation of inequality true.
A number is a solution of an equation or inequality if
that number makes the equation or inequality true.
For example, the number x=0.5 is a solution of x%5E2-1%3C0,
because 0.5%5E2-1=0.25-1=-0.75%3C0 ,
and the set of numbers in the interval (-1,1) is the solution set for x%5E2-1%3C0 ,
because all the numbers in that interval make x%5E2-1%3C0 true, and no other numbers are solutions.
x=1 and x=-1 are solutions for x%5E2-1%3C0 ,
because they make x%5E2-1%3C0 true.
The set {-1,1} is the solution set for x%5E2-1%3C0
because x=-1 and x=1 are the only solutions to x%5E2-1%3C0 .
x=1 and x=-1 are solutions for x%5E2-1%3C0 ,
because they make x%5E2-1%3C0 true.