Quadratic Inequalities

• A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c > 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$. The inequality symbol can also be $<$, $\leq$, or $\geq$.

• Unlike quadratic equations which yield specific points, inequalities describe intervals on the number line where the quadratic function is either above or below the x-axis.

• The critical values of the inequality are the roots of the corresponding quadratic equation $ax^2 + bx + c = 0$. These values mark the boundaries where the expression changes sign.

The Four-Step Method

• Step 1: Rearrange: Move all terms to one side to obtain a quadratic expression compared to zero ($ax^2 + bx + c > 0$). Ensure the $x^2$ coefficient is positive by multiplying by $-1$ and flipping the inequality sign if necessary.
• Step 2: Find Roots: Solve the equation $ax^2 + bx + c = 0$ using factoring, completing the square, or the quadratic formula to find the critical values $x_1$ and $x_2$.
• Step 3: Sketch: Draw a quick sketch of the parabola. If $a > 0$, it is a 'valley' shape; if $a < 0$, it is a 'hill' shape.
• Step 4: Identify Region: Look at the inequality sign. If it is $> 0$, choose the regions where the curve is above the x-axis. If it is $< 0$, choose the region where the curve is below the x-axis.

• It is vital to distinguish between strict inequalities ($<, >$) and non-strict inequalities ($\leq, \geq$). Strict inequalities use open circles on a number line and round brackets in interval notation, while non-strict use solid circles and square brackets.

• Algebraic vs. Graphical: While one can use test points in each region, sketching the parabola is generally faster and less prone to arithmetic errors.

• Always Sketch: Examiners often look for a sketch to award method marks. It serves as a visual proof of your logic and prevents 'inside-out' interval errors.

• Check the $x^2$ term: If you start with a negative $x^2$ term, it is often safer to multiply the whole inequality by $-1$ and flip the sign. This ensures you are always dealing with a U-shaped parabola, making the 'between' or 'outside' rule consistent.

• Verify Endpoints: If the inequality is $\leq$ or $\geq$, ensure your final answer includes the equals sign. If the original question has no equals sign, your answer must not have them either.

• Dividing by $x$: Never divide an inequality by a variable like $x$. Since $x$ could be negative, you wouldn't know whether to flip the inequality sign, and you might lose a critical root at $x=0$.

• The 'Single Inequality' Error: Students often try to write $x < x_1 > x_2$ as a single statement. This is mathematically incorrect. Use the word 'or' for disjoint intervals.

• Ignoring the Sign Flip: Forgetting to reverse the inequality sign when multiplying or dividing by a negative number is the most frequent source of lost marks.