Solving Polynomial Inequalities
📘 Key Definitions
- Polynomial Inequality
- An inequality where both sides are polynomials. After simplification, it takes the form $P(x) > 0$, $P(x) < 0$, $P(x) \geq 0$, or $P(x) \leq 0$.
- Degree of the Inequality
- The degree of the resulting polynomial $P(x)$ when all terms are brought to one side and combined in standard form.
- Solution Set
- The set of all real numbers $x$ that make the inequality true, usually expressed as a union of intervals.
1️⃣ First-Degree (Linear) Inequalities
A first-degree polynomial inequality has the form:
$$ax + b > 0 \quad \text{(or } <, \geq, \leq \text{)}$$
The solution depends on the sign of the leading coefficient $a$:
If $a > 0$:
$$ax + b > 0 \implies x > -\frac{b}{a}$$
$$ax + b < 0 \implies x < -\frac{b}{a}$$
If $a < 0$:
$$ax + b > 0 \implies x < -\frac{b}{a}$$
$$ax + b < 0 \implies x > -\frac{b}{a}$$
⚠️ The inequality flips when dividing by a negative number!