Equations and Inequalities
What the exam tests
Equations and Inequalities tests equivalent transformations: solving linear and quadratic equations, systems, polynomial and rational inequalities, and absolute-value statements. Kyoto problems often ask for a parameter range or for the exact set of solutions, so endpoint behavior matters.
Core rules
- An equation is transformed equivalently by adding the same expression or multiplying by a nonzero constant. Multiplying by an expression containing the variable can introduce extraneous roots.
- For ax^2 + bx + c = 0, the discriminant D = b^2 - 4ac determines the number of real roots; D > 0 gives two, D = 0 one, D < 0 none.
- Solve polynomial inequalities by finding all real zeros and testing the sign on each interval, or by using a sign chart.
- |A| < B is equivalent to -B < A < B only when B > 0. More generally, |A| < B requires B ≥ 0 and gives -B < A < B.
Worked example
Solve |x - 3| < 2x. If x < 0, the right side is negative while the left side is nonnegative, so no solution; hence x ≥ 0. Both sides are now nonnegative, so squaring is reversible: (x - 3)^2 < 4x^2, giving x^2 - 6x + 9 < 4x^2, or 3x^2 + 6x - 9 > 0. Divide by 3: x^2 + 2x - 3 > 0, so (x + 3)(x - 1) > 0. This gives x < -3 or x > 1. Intersecting with x ≥ 0 leaves x > 1.
Traps
Do not multiply an inequality by x without splitting cases by its sign. Squaring is safe only when both sides are nonnegative. Check whether endpoints are included and reject solutions that make a denominator zero.