Why do I flip the inequality sign?

When you multiply or divide both sides by a negative number, the direction of the inequality reverses. For example, −2x > 4 becomes x < −2. You do NOT flip the sign when adding, subtracting, or multiplying/dividing by a positive number.

How do I solve a quadratic inequality?

Factorize and find the roots α ≤ β. If the leading coefficient is positive, the expression is positive outside the roots (x β). If negative, multiply the entire inequality by −1 (flipping the sign) to make the x² coefficient positive first, then apply the inside/outside shortcut.

What is the Vertex Theorem in Linear Programming?

The optimal (maximum or minimum) value always occurs at one of the vertices of the feasible region. Find all corner vertices by solving simultaneous equations of intersecting boundary lines, then test each vertex's coordinates in the objective function to find the maximum or minimum value.

Inequalities & Linear Programming

Q: How do I solve a quadratic inequality?

Factorize and find the roots α ≤ β. If the leading coefficient is positive, the expression is positive outside the roots (x β). If negative, multiply the entire inequality by −1 (flipping the sign) to make the x² coefficient positive first, then apply the inside/outside shortcut.

Master compound linear inequalities, solve quadratic inequalities, and learn how to construct bounded feasible regions to find optimal real-world solutions under constraints.

1. The Core Concept

The Geometry of Constraints: Unlike equations which give specific values, solving an inequality gives a range of valid values. Geometrically, this range can be represented as a shaded segment on a number line, or as a bounded region (feasible region) on a coordinate plane in Linear Programming.

2. DSE Inequalities & Linear Programming Cheat Sheet — 5 Key Topics

Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.

Topic	Key Concept & Steps	Exam Shortcuts & Tips
1. Linear & Compound	• General Form: `ax > b` • AND: Overlapping region on a number line. • OR: Combined union of both regions.	The Golden Rule: Always flip the inequality sign when multiplying or dividing by a negative number. e.g., `−2x > 6 ⟹ x < −3` Draw a number line diagram to find the overlap for "AND" or the union for "OR" compound systems.
2. Quadratic	For `x² + bx + c > 0` or `< 0` with roots `r₁ < r₂`: • If `> 0`: Outside range (`x < r₁` or `x > r₂`) • If `< 0`: Inside range (`r₁ < x < r₂`)	Find the roots of the quadratic equation first. Then apply the "inside/outside" shortcut. Ensure the x² coefficient is positive before applying the "inside/outside" shortcut.
3. 2D Linear Inequalities	`Ax + By ≤ C` or `Ax + By > C`	Solid vs. Dotted Boundaries: • Solid line for `≤` and `≥` (boundary included). • Dotted line for `<` and `>` (boundary excluded). Pick a test point (usually `(0,0)`) to determine which side of the boundary line to shade.
4. Feasible Region	The overlapping shaded region formed by a system of multiple 2D linear inequalities.	Graph all boundary lines, shade their common intersection area, and identify all corner vertices. Locate all corner vertices (corners) by solving simultaneous equations for where the boundary lines intersect.
5. Linear Programming	Optimize (maximize or minimize) the Objective Function: `P = ax + by`	The Vertex Theorem: The optimal (maximum or minimum) value always occurs at one of the vertices of the feasible region. Test the coordinates of each corner vertex in the objective function to find the maximum or minimum value.

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💡 Exam Traps & Secrets

⚠️ The Negative Sign Flip TrapThis is the single most common careless error in Section A1. You only flip the inequality sign when multiplying or dividing by a negative number. Do not flip the sign if you are just subtracting or adding a negative number!
Subtraction (No Flip): x − 3 < 5 ⟹ x < 8
Division (Must Flip): −2x > 6 ⟹ x < −3

⚠️ The "AND" vs. "OR" Overlap Rule"AND": The final solution is only where the shaded lines overlap on the number line. If the two arrows point away from each other and do not overlap, the system has no solution.
"OR": The final solution is any region covered by at least one of the arrows.

⚠️ Linear Programming Boundary Line StylesThis is heavily tested in Paper 2. Inclusive signs (≤, ≥) mean solid lines. Strict signs (<, >) mean dotted/dashed lines.

⚠️ Objective Function Slopes (Paper 2 Shortcut)If you need to quickly identify which vertex is optimal in Paper 2 without testing all 4 points, rearrange the objective function P = ax + by into slope form:
y = −(a/b)x + P/b
By shifting this parallel slope line across the feasible region, you can visually spot which corner vertex it touches last (maximum) or first (minimum).

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Compound "AND" compound boundaries)Solve the simultaneous inequalities 3x − 7 < 2x + 1 and 5 − 2x ≤ 11, and express your answer on a number line.

(Quadratic "outside range" shortcut)Solve the quadratic inequality x² − 5x + 6 > 0.

(Linear Programming optimization using the Vertex Theorem)Graph the feasible region for x ≥ 0, y ≥ 0, and x + y ≤ 5, and find the maximum value of the objective function P = 3x + 2y.

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