Before diving into equations, we need to be comfortable with the building blocks: real numbers. These are the numbers you see on a number line, from negatives to positives, including fractions and decimals. When we work with them, they follow a few consistent rules that make algebra predictable.

The commutative property means order doesn't matter. The associative property means you can regroup numbers without changing the result. The distributive property is how multiplication interacts with addition or subtraction inside parentheses. You'll use it constantly to simplify expressions.

To keep our calculations consistent, we follow a specific order of operations. You might know it as PEMDAS.

An equation is a statement that two things are equal. A linear equation is one where the variable isn't raised to any power higher than one. Think of it as a balanced scale. Whatever you do to one side, you must do to the other to keep it balanced.

The goal is always to isolate the variable, to get it by itself on one side of the equals sign. We do this by performing inverse operations. The inverse of addition is subtraction, and the inverse of multiplication is division.

Inequalities work almost exactly the same way. The only difference is that instead of an equals sign, you have symbols like $<$ (less than), $>$ (greater than), $\le$ (less than or equal to), or $\ge$ (greater than or equal to).

There's one crucial rule to remember: if you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign.

A function is a rule that assigns exactly one output for each given input. Think of it like a machine. You put something in (the input), the machine does something to it, and something new comes out (the output). For any specific input you use, you will always get the same output.

We call the set of all possible inputs the domain, and the set of all possible outputs the range. We often write functions using a special notation, like $f(x) = 2x + 1$. This is read as "f of x equals 2x plus 1." The $f(x)$ part is just another way of writing the output variable, usually $y$.

To evaluate a function, you just replace the variable with the given input value. For example, to find $f(3)$ for the function $f(x) = 2x + 1$, we substitute 3 for $x$:

$f(3) = 2(3) + 1 = 6 + 1 = 7$.

So, for an input of 3, the output is 7.

A polynomial is an expression made of variables, constants, and exponents, combined using addition, subtraction, and multiplication. The exponents must be non-negative whole numbers.

We classify polynomials by their number of terms. A monomial has one term ($5x^2$), a binomial has two terms ($3x - 1$), and a trinomial has three terms ($x^2 + 4x + 4$). The degree of a polynomial is the highest exponent of the variable.

To add or subtract polynomials, you simply combine like terms. Like terms are terms that have the exact same variable part, including the exponent. You just add or subtract their coefficients.

Multiplying polynomials involves using the distributive property. You must multiply every term in the first polynomial by every term in the second one. When multiplying binomials, a common method is FOIL.

Ready to check your understanding? Let's try a few questions.

These foundational concepts are the tools you'll use to solve more complex problems.