Arithmetic expression: numbers and operators. PEMDAS order. No variables.
Ex.    5+6-2(32-√25)     which simplifies / evaluates / calculates / figure-out-what-it-equals / "boils down"    to 3


An arithmetic expression is a (restricted) algebraic expression.

       

Exs:
   6x
   2x + 5
   7x2 + 3x + 4
   3x2yz3 + y - 4x
   -1.23√(2x+5) + 4
   (7x2+3x+4) / (2x+5)
   3·3.212x+5 + 4
   3·log(2x+5) + 4
   3·sin(2x+5) + 4

Variable is a placeholder. Letters are used to indicate variables. Could have been boxy symbols:
7▢2+3▢+4    instead of   7x2+3x+4   to convey the idea that the variable can be or take on any value (fill in the box).
3▢-5△+4   instead of   3x-5y+4

In an application the variables could represent some quantity of stuff such as time, money, energy, mass, temperature, length, area, volume, etc. Typically, a more mnemonic letter is used, e.g. t for time.
Ex.   5/9(F-32)    F is temperature in fahrenheit
Ex.   ½bh    b is base, h is height of a triangle
Ex.   P(1+r)t    P is the principle (lump of money), r is the interest rate, t is years.
Ex.   -4.9t2+vt    t is time in seconds, v is velocity in m/s

What can be done with an algebraic expression:

  1. Simplify it by combining like terms, if needed or wanted e.g. to minimize subsequent evaluation or to clarify what's going on.
    Ex.    3x2+5x+4x2-2x+4 → 7x2+3x+4

  2. Factor it, especially a polynomial, to break it down into smallest factors (pieces, components, building blocks).
    Ex.    7x2+3x-4 factors into (7x-4)(x+1)
    Ex.    3x3-75x factors into 3x(x-5)(x+5)

  3. Determine its domain (the set of values the variable(s) can possibly be).
    Basically, denominators can not be zero and can not do square root of a negative number.
    Ex.    1/x    x can be any real number except 0
    Ex.    1/(x-2)2    x can be any real number except 2
    Ex.    (2x+1)/(2x2+2x-4)    x can be any real number except -2 and 1 (factor the denominator)
    Ex.    √(3x+4)    x can be any real number ≥ -4/3.   [-4/3,∞)
    Graph these to see what can happen at numbers not in the domain.

  4. Evaluate it by substituting a value (number) for each different variable and then calculating the resulting arithmetic expression.
    Ex.    Evaluate 7x2+3x+4
    when x is 3 → 7(3)2+3(3)+4 = 76
    when x is 0 → 7(0)2+3(0)+4 = 4
         0 is the "best" number to evaluate with because it makes every term containing the variable zero i.e. "disappear".
         And 0 often means starting value, nothing, no time, no money, etc.
    when x is 1 → 7(1)2+3(1)+4 = 14
    when x is 1.234 → 7(1.234)2+3(1.234)+4 = 18.361292

  5. Each side of an equation is an algebraic expression.

    Equation of one variable:
    Ex.    7x2+2x+4 = 3x+12
    can be solved, i.e. find what value(s) for x makes the equation true, if any. This one has two solutions, x= 8/7 and -1.
    Substituting 8/7 for x in both sides of the equation results in the True equation 15.428571=15.428571.
    Substituting -1 for x in both sides of the equation results in the True equation 9 = 9.
    Every other number results in a False equation. We want to know the Truths.
    NB. an equation of one variable does not have a graph.

    An equation in two variables, say x and y, can be graphed in the Cartesian rectangular coordinate plane, each point of the graph being a solution to the equation. A solution is an x and y pair of numbers that when substituted into the equation's expressions evaluates to a True equation. The graph is the set of all solutions, it is a "picture" of the equation's solution set. NB. an expression does not have a graph, pace Desmos.
    Ex.    3y2-x2 = 3x-2y   
         has an infinite number of solutions, (0,0), (0,-2/3), (-3,0) being three of them

    If the two-variable equation can be solved for y (i.e. gotten into the form y = expression_in_x), then that expression can be the definition of a function of x [see below].
    Ex.    y3-x2 = 3x+2 can be solved for y:    y = ∛(x2+3x+2)

    A formula is a "well-known" equation:
    A = ½bh
    C = 5/9(F-32)
    A = πr2
    A = P(1+r)t
    E = mc2

  6. An algebraic expression can be the definition of a function.
    Ex.    ƒ(x) = 7x2+3x+4,    a quadratic function.
    Ex.    ƒ(x) = 3x+4,    a linear function.
    Ex.    ƒ(x) = ∛(x2+3x+2),    the above example.

    A function can be evaluated at different values of the variable (x inputs), resulting in the value of the function (y outputs). The graph of the function is the set of these (x,y) pairs. The function values ƒ(x) are the y's, so y=ƒ(x).
    A function is a restricted equation of the form y = one_expression_in_x

  7. Each side of an inequality is an algebraic expression.
    Ex.    7x2+2x+4 ≤ 3x+12
    An inequality can be solved, i.e. what values when substituted into the variable(s) yields Truth. Typically an inequality's solution set is an interval; the example's solution set is: −1≤x≤8/7, or [-1,8/7], or graphically: