First and second degree inequalities

Linear inequalities

You solve $ax + b > 0$ like an equation, with one extra rule: multiplying or dividing both sides by a negative number reverses the inequality. The reason is that multiplying by $-1$ reflects the number line: $2 < 5$, but $-2 > -5$.

Adding or subtracting anything, and multiplying or dividing by a positive number, keeps the direction. So $-3x > 6$ gives $x < -2$, while $3x > 6$ gives $x > 2$.

The answer is a set of numbers, usually an interval: $x < -2$ is the interval $(-\infty, -2)$; $-1 \le x < 4$ is $[-1, 4)$. A square bracket includes the endpoint, a round one excludes it.

A double inequality such as $-1 < 2x + 3 \le 7$ is solved on all three parts at once: subtract 3 to get $-4 < 2x \le 4$, then divide by 2 to get $-2 < x \le 2$.

Quadratic inequalities: think of the parabola

To solve $ax^2 + bx + c > 0$ (or $<$, $\ge$, $\le$), do not try to "take square roots". Instead:

1. Find the roots of $ax^2 + bx + c = 0$.
2. Picture the parabola $y = ax^2 + bx + c$. If $a > 0$ it opens upwards: it is negative between the roots and positive outside them. If $a < 0$ it is the other way round. (Or multiply by $-1$ first to make $a$ positive, flipping the sign.)
3. Read off the answer.

For example, $x^2 - x - 6 = (x + 2)(x - 3)$ has roots $-2$ and $3$, and opens upwards, so:

Inequality	Solution
$x^2 - x - 6 < 0$	$-2 < x < 3$
$x^2 - x - 6 \le 0$	$-2 \le x \le 3$
$x^2 - x - 6 > 0$	$x < -2$ or $x > 3$
$x^2 - x - 6 \ge 0$	$x \le -2$ or $x \ge 3$

When $\Delta < 0$ there are no roots and the parabola never crosses the axis: with $a > 0$ it is always positive. So $x^2 + 2x + 5 > 0$ holds for every real $x$, and $x^2 + 2x + 5 < 0$ for none. When $\Delta = 0$, the parabola touches the axis once: $(x - 1)^2 > 0$ holds for every $x$ except $x = 1$, and $(x - 1)^2 \le 0$ only at $x = 1$.

A common special case: $x^2 > 9$ means $x < -3$ or $x > 3$, not just $x > 3$; and $x^2 < 9$ means $-3 < x < 3$.

Fractional inequalities: the sign chart

For $\frac{N(x)}{D(x)} \ge 0$, you may not multiply both sides by $D(x)$, because you do not know whether it is positive or negative, so you do not know whether to flip. Instead:

1. Bring everything to one side and combine into one fraction, so the other side is $0$.
2. Find the zeros of the numerator and of the denominator.
3. Mark them on a number line and record the sign of each factor in each interval.
4. Choose the intervals with the sign you want. A zero of the numerator is included for $\ge$ or $\le$; a zero of the denominator is never included.

The same chart solves products: $(x - 1)(x + 4)(x - 6) < 0$ changes sign at each root.

Check with a test value

Whatever the method, pick a number inside your answer and one outside and substitute them. It takes seconds and catches a reversed sign every time.