Distinction between problems (such as equations), and universal truths

Question

How the distinction between problems (find/describe such values of x that… ) and universal truths (identities) is taught to secondary-school students and higher? Especially in English-speaking countries.

Ī mean the following thing (sorry if Ī present an obvious stuff in a confused way): if we have a relation, equality or else, and a proposition with this relation at the top and some variable(s), then we have two distinct useful cases, although notation might be the same. In the first case the proposition is not necessarily true and we have to find values of the variable where it is true. In the second case the proposition is necessarily true (law/identity/“theorem”) and variables become bound by universal quantification. Universal truths can be used in subsequent equivalent transformations of expressions, equations, inequalities, or something alike. Students must learn to distinguish these cases, haven’t they? Examples:

Binary relations:         “=”  (equals)     “>” (greater than)

   Problems:   “$x^2 - x - 1 = 0$” (equation)   “$x^2 - x - 1 > 0$”

Universal truths:   “$(a+b)(a-b) = a^2 - b^2$”    “$\exp(x) > 0$”

How may be used: “α² = δ² ⇒ α = δ ∨ α = −δ ”      exp(α)⋅ℓ > 0 ⇔ ℓ > 0
          (transformed equation)       (transformed inequality (problem))
          “(α + 1)(α − 1) = α² − 1 ”       cosh α > 0
          (transformed expression)      (derived inequality (truth))

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Why am Ī preoccupied with it? A year ago, the only person happened to cooperate with me about precise equation–identity distinction in English Wikipedia was a mathematician from France; was a coincidence he wasn’t taught in English? Today Ī argued on this topic (against two presumedly native English speakers) at a neighbouring site and become frustrated. Could an English speaker be correct in insisting that something like aⁿ⁺¹ = a × aⁿ is an “equation” and calling the “=” symbol in it “equation sign”?

Answer 1

I am comfortable saying "Solve the equation $x+2$=4" and also saying "Using the equation $(a+b)(a-b)=a^2-b^2$, we see that...".

On other other hand I would only ever speak of solving an equation, not an identity, and I would be willing to use the word "identity" in my second example above.

So I think I would say that an identity is a kind of equation, one which is universally quantified.

In general, one can avoid confusion jut by saying what you mean, as in "Solve the equation $x+2=4$", or "For all numbers $a$ and $b$, $(a+b)(a-b)=a^2-b^2$"

Answer 2

In my experience (being taught in a US school), there is no explicit distinction between an identity and an equation. The reason being is that to make the distinction you have to introduce first order logic which is not generally appropriate for secondary (high school) mathematics education. Students can usually ascertain an identity (a formula) from an equation to be solved so this is not usually an issue. The students that have trouble using formulas or have trouble solving equations are unlikely to benefit from a more detailed explanation (the detailed explanation will likely only further confuse them and make the topic of mathematics far less tangible and far more abstract).

Examples:

1. Quadratic equation.

We are generally taught that given $ax^2 + bx + c = 0$ such that $a \neq 0$ that $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This is an identity. It's a first order equation. Formally it would state the following:

$$ \forall x \in \mathbb{C}.\forall a, b, c\in \mathbb{R}:\left( \left(a\neq 0\wedge ax^2 + bx + c = 0\right) \rightarrow x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\right) $$

2. Solving an equation:

Solve the following equation: $10x^2 + 6x + 8 = 0$ wher $x$ must be a real number.

The very first thing to do when solving an equation is to determine whether or not it can be solved. This too is a first order logic equation. The question (which must be answered) is whether or not the following is true:

$$ \exists x\in \mathbb{R}: 10x^2 + 6x + 8 = 0 $$

In this particular instance, the answer would be that the above is false therefore there are no solutions. But how do you know that? How do you prove that?

Here is a different example: Solve the following: $x^2 + 6x + 8 = 0$. Again, we must first decide whether or not the following first order equation is true or false:

$$ \exists x\in \mathbb{R}: x^2 + 6x + 8 = 0 $$

In this case it is true (how do we know?). Once we know the equation can be satisfied, we next attempt to find the particular solutions. This is not easy and in fact, "impossible" in general. Here is an example:

Find real values of $x$ that satisfy the following equation

$$ e^x = x + 2 $$

It can be proved that a solution exists:

$$ f(x) = e^x - x - 2 \rightarrow \text{find } f(x) = 0 \\ f'(x) = e^x - 1 \rightarrow f'(x) = 0 \rightarrow e^x = 1 \rightarrow x = \ln(1) = 0 $$

Since the derivative changes from negative to positive at $x = 0$, this represents a relative minimum (and since this is the only extrema this is the global minimum). $f(0) = 1 - 0 - 2 = -1$. Further since $\lim_{x\rightarrow-\infty} e^x = 0$ and $\lim_{x\rightarrow-\infty} = -\infty$ (thus subtracting negative infinity creates a positive infinity) and that $e^x$ dominates $-x$ such that $\lim_{x\rightarrow\infty} e^x - x = +\infty$, we know that this function, $f(x)$, goes towards $+\infty$ on either side, there must be two spots where $f(x) = 0$ and therefore there are exactly two real solutions to $e^x = x + 2$.

Yet even though we know the equation to be satisfiable, it is not clear how (or analytically possible) to solve for those two values.

So when solving equations there are two separate issues that we usually do not identify in secondary education (largely because it's beyond the scope). The first is, whether or not there is a solution and the second is how to find the solution once you can show that one exists--note that the first step becomes tedious to the point of being prohibitive towards learning for the vast majority of problems encountered in elementary to high school level mathematics courses.