Inequalities – Lecture Review

Flashcards covering core concepts from the lecture on solving and graphing inequalities, including sign changes, quadratic inequalities, set notation, and graphical conventions.

What happens to the inequality sign when you multiply or divide an inequality by a negative number?

The direction of the inequality sign flips (e.g., −3x < 6 becomes x > −2 after dividing by −3).

Does adding or subtracting the same number on both sides of an inequality change its direction?

No. Adding or subtracting the same value on both sides leaves the inequality direction unchanged.

Outline the steps for solving the quadratic inequality 36x ≤ 6x².

Rearrange to 6x² − 36x ≥ 0, factorise as 6x(x − 6) ≥ 0, then use a sign/graph check to get the solution x ≤ 0 or x ≥ 6.

Why should you avoid dividing an inequality by a variable whose sign is unknown?

If the variable is negative, dividing by it would reverse the inequality sign, potentially giving an incorrect result.

How can sketching a graph help you solve a quadratic inequality?

By plotting the curve, you can visually see which x-values lie above or below the x-axis (or another curve) and thus satisfy the inequality.

Which symbols are used in set notation to denote union and intersection?

Union is represented by U, and intersection is represented by ∩ (often written as 'n').

How do you test which side of a boundary curve is the valid region for an inequality?

Choose a test point (commonly the origin), substitute it into the inequality, and see if it satisfies the inequality; if yes, that side is included.

When graphing y ≤ −x² + 2x + 3 and y ≥ 1, how do you identify the solution region R?

Sketch both curves, test a sample point (e.g., the origin) against each inequality, and shade the area where both conditions overlap.

What is the graphical difference between dashed and solid boundary lines when plotting inequalities?

Solid lines are used for ≤ or ≥ (inclusive boundaries); dashed lines are used for < or > (exclusive boundaries).

In an exam, what key conventions should you follow when representing inequalities graphically?

Use correct line style (solid/dashed), clearly label curves/axes, test a point to identify the solution region, and express the answer clearly, often in set notation.