Linear, Quadratic, Fractional and Absolute Value Inequalities

• Solve as for a linear equation, but reverse the sign whenever you multiply or divide both sides by a negative number.
• A system (both conditions must hold) is solved one inequality at a time, then the solution sets are intersected on a number line.
• Conditions joined by or give the union of the solution sets.
• An empty intersection means no solution ($\varnothing$); an inequality that reduces to a true statement such as $0 < 4$ holds for every real $x$ ($\mathbb{R}$).

• Move everything to one side so the other side is 0. If $a < 0$, multiply by $-1$ and reverse the sign.
• With $a > 0$ and roots $x_1 < x_2$: the expression is negative between the roots and positive outside them.
• $\Delta < 0$: always positive. So $> 0$ holds for all of $\mathbb{R}$, and $< 0$ has no solution.
• $\Delta = 0$: $a(x - r)^2$ is positive except at $x = r$, where it is 0.
• Include the roots for $\le$ and $\ge$; exclude them for $<$ and $>$.

• Move everything to one side and combine into a single fraction $\frac{N(x)}{D(x)}$ compared with 0.
• Find where $N = 0$ and where $D = 0$; mark them on a number line and record the sign of each factor in each region.
• The fraction is positive where $N$ and $D$ have the same sign, negative where they differ.
• Zeros of $D$ are always excluded. Zeros of $N$ are included only for $\le$ or $\ge$.
• A factor that is always positive (such as $x^2 + 1$) never changes the sign: ignore it.

For $k > 0$:

• $|A| < k$ means $-k < A < k$: one interval.
• $|A| > k$ means $A < -k$ or $A > k$: two intervals.
• For $k < 0$: $|A| < k$ has no solution and $|A| > k$ holds wherever $A$ is defined.
• $|A| \ge 0$ always, and $|A| = 0$ only where $A = 0$. So in a fraction, a numerator like $|A| + 1$ is always positive and only the denominator decides the sign.