Inequalities

An inequality is a mathematical statement that compares two expressions using symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).

1. Linear Inequalities

Linear inequalities involve variables raised to the power of 1. Solving them is similar to solving linear equations, with one crucial rule: If you multiply or divide both sides by a negative number, you must reverse the inequality sign.

• Analytical Solution: Isolate the variable using algebraic operations. Example: 3x - 5 > 7 → 3x > 12 → x > 4.
• Graphical Solution: On a number line, use an open circle for < or > and a solid circle for ≤ or ≥. For inequalities in two variables (e.g., y > x + 1), the solution is a shaded region on a Cartesian plane.

2. Quadratic Inequalities

Quadratic inequalities involve terms with x². In the JAMB syllabus, these typically feature integral roots (whole numbers).

Steps to Solve Quadratic Inequalities:

• Factorization: Arrange the inequality in the form ax² + bx + c > 0 (or < 0) and factorize the quadratic expression.
• Find Critical Points: Set the factors to zero to find the roots (boundary values).
• Testing Intervals: Use a number line or a table of signs to determine which regions satisfy the inequality.
• Example: x² - 5x + 6 < 0 → (x - 2)(x - 3) < 0. Testing values between 2 and 3 shows the expression is negative. Solution: 2 < x < 3.

3. Graphical Interpretation

When looking at graphs:

• A dashed line indicates that the boundary is not included (< or >).
• A solid line indicates that the boundary is included (≤ or ≥).
• The shaded region represents the set of all points (x, y) that satisfy the inequality.