Inequalities
Inequalities
Solve inequalities by transforming them into equivalent forms, tracking the domain and the direction of the inequality. For quadratics, factor or use a sign chart; for rational inequalities, bring everything to one side and use critical points, never multiply blindly by an expression whose sign is unknown. Absolute values split into cases or mean distance on the number line. Know the basic inequalities: for positive a and b, a²+b²≥2ab and (a+b)/2≥√(ab), with equality exactly when a=b. These give maximum and minimum values when the sum or product is fixed.
Traps: multiplying or dividing by a negative quantity without reversing the sign; squaring both sides when one side may be negative; forgetting that AM–GM requires positive terms; and quoting equality without checking whether it can occur. Also check endpoints and whether the inequality is strict, and do not confuse the solution of f(x)>0 with the range of f.
Worked example: solve |2x−1|<3. This means −3<2x−1<3, so −2<2x<4 and −1<x<2. For a minimum problem, x+4/x for x>0 has minimum 4 at x=2 by AM–GM, since x+4/x≥2√(x·4/x)=4, with equality when x=2. Rule of thumb: record the domain first, then use a sign chart or a standard inequality, and state the equality condition.