Linear Inequalities

A linear inequality is simply a linear equation with the equals sign swapped for $<$, $>$, $\le$ or $\ge$, so instead of "what number works?" it asks "which numbers work?" Think of a budget: you don't have to spend exactly 100, just at most 100. The rule up top is the one new thing to learn: $a < b$ (one number is smaller), $c < 0$ (you multiply by a negative), and then $ac > bc$ (the order flips). Everything else is the same as solving equations.

Why the flip? Look at $2 < 3$. Multiply both by $-1$ and you get $-2$ and $-3$, but now $-2$ is the bigger one, so $-2 > -3$. Multiplying by a negative mirrors the number line around zero, and a mirror reverses left and right. Adding, subtracting, or multiplying by a positive never does that, so those moves leave the sign alone.

Interval notation. Since the answer is a range, we need a way to write ranges. $[a, b)$ means everything from $a$ to $b$, including $a$ but not $b$. A square bracket means the endpoint is included, a parenthesis means it isn't, and $\pm\infty$ always gets a parenthesis, since infinity isn't a number you can land on.

Double inequalities like $-3 < 2x + 1 \le 7$: do the same operation to all three parts at once, and the middle ends up as just $x$.

Budgets, dosage limits and safety tolerances are inequalities: spend at most this, stay above that.