Question

how to determine whether an equation is always, sometimes , or never true?

Answer 1

Given some equation, if you can find a single solution, then the equation is sometimes true. "Sometimes" true because it's only true for this one solution.

For example, \(x+2=2\) is sometimes true, since it's only true for \(x=0\).

If, when you try to solve, you end up with an identity, like \(1=1\) or \(0=0\), then the equation is always true.

For example, \((x-2)^2=x^2-4x+4\):
\[\begin{align*}(x-2)^2&=x^2-4x+4\\
x^2-4x+4&=x^2-4x+4\\
0=0\end{align*}\]
So no matter what you plug in for \(x\), you end up with \(0=0\), which means the equation is always true.

An equation is never true if you end up with something that's, well, not true.

For example, \(x+2=x+3\) gives you \(2=3\), which is not true.

Answer 2

THANK YOU !!!!

Answer 3

You're welcome!