Equation

An equation is a claim, not an instruction

$$x + 3 = 10$$

This does not tell you to do anything. It asserts that the quantity on the left and the quantity on the right are the same number. Solving asks: for which values of $x$ is that assertion true?

Reading the equals sign as “and the answer is” is a habit from arithmetic that causes trouble in algebra, because in $3x + 7 = 22$ nothing is being computed left to right.

Equation, expression, inequality

	Has an equals sign?	Can be solved?
Expression, e.g. $3x + 7$	no	no — it can only be simplified or evaluated
Equation, e.g. $3x + 7 = 22$	yes	yes
Inequality, e.g. $3x + 7 < 22$	no, uses $<$ or $>$	yes, giving a range

An expression has no solution because it makes no claim. Asking to “solve $3x + 7$” is asking a question that has not been posed.

Keeping an equation true

Any operation applied to both sides preserves the truth of the claim, which is what makes inverse operations safe. Change one side alone and you are looking at a different equation with, in general, a different solution set.