🚦 No Crossing the Line: Conquering Inequalities!

Key Concepts

• Inequality symbols $<,\;>,\;\le,\;\ge$ describe a *range* of values. Treat them like equations **except** that multiplying or dividing by a negative flips the sign.
• For a quadratic inequality, factor (or use the quadratic formula), find critical values, sketch the parabola, then read off where it sits above or below the $x$-axis.

Worked Example: Between the Roots

• Problem: Solve the inequality $x^2 - 5x + 6 < 0$.
• Step 1:

  $$x^2 - 5x + 6 = (x-2)(x-3)$$
• Step 2: Critical values occur at $x=2$ and $x=3$. Since the coefficient of $x^2$ is positive, the parabola opens upwards.
• Step 3: The graph is below the $x$-axis **between** the roots, giving $2 < x < 3$. **Practise** similar problems to boost speed and confidence!