Why can't I just solve it like an equation?

Because the answer is a range of values, not two numbers. The roots are only the boundaries; the sketch tells you which side of them is the answer.

When do I write 'or' and when do I write two inequalities together?

Between the roots it is one statement, like −3 4, which can never be written as one.

How to Solve Quadratic Inequalities Step by Step

Short answer

How do you solve a quadratic inequality?

Move everything to one side, solve the equation to find the two roots, then sketch the parabola. For a positive x² term, the quadratic is below zero between the roots and above zero outside them.

A quadratic inequality asks where a quadratic is positive or negative. The answer is a range of $x$ values.

The method

Rearrange so one side is 0.
Solve the equation to find the critical values (the roots).
Sketch the parabola through those roots and read off the answer.

For a positive $x^{2}$ coefficient the parabola is a U shape: it is below the axis between the roots and above it outside them.

Example 1: greater than zero

Solve $x^{2} - 2 x - 8 > 0$ .

x^{2} - 2 x - 8 = (x - 4) (x + 2) = 0 ⟹ x = 4 or x = - 2

We want where the U shape is above the axis, which is outside the roots:

x < - 2 or x > 4

Example 2: less than zero

Solve $2 x^{2} - 9 x - 5 < 0$ .

(2 x + 1) (x - 5) = 0 ⟹ x = - \frac{1}{2} or x = 5

Below the axis means between the roots:

- \frac{1}{2} < x < 5

Example 3: the one everyone gets wrong

Solve $x^{2} \leq 9$ . Writing $x \leq \pm 3$ is wrong. Rearrange to $x^{2} - 9 \leq 0$ , so the roots are $\pm 3$ and the answer is between them:

- 3 \leq x \leq 3

Try it: test the sign on each interval

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Mistakes that cost marks

Stopping at the roots. The roots are boundaries, not the answer.
Picking the wrong side. Sketch the curve every time; it takes five seconds.
Using $<$ where the question says $\leq$ . A $\leq$ includes the roots themselves.

How to solve quadratic inequalities