Equations and Inequalities
In the equations and inequalities domain, Bagrut questions test equivalence and solution sets rather than isolated tricks. Expect linear and quadratic equations, systems, rational and radical equations, absolute value, and inequalities solved with sign tables or graphs.
Core rules
- A quadratic ax²+bx+c=0 has discriminant Δ=b²−4ac; Δ>0 gives two real roots, Δ=0 one repeated root, Δ<0 no real roots.
- Vieta: sum of roots = −b/a, product = c/a.
- Squaring both sides can create extraneous roots, so check every candidate in the original equation.
- |x−a|<d means a−d<x<a+d; |x−a|>d splits into two rays.
- For a rational inequality, move everything to one side, factor, and use a sign table; never multiply by an expression whose sign is unknown.
- A system is solved by substitution or elimination; the solution must satisfy every equation.
Worked example
Solve (x−1)/(x+2) ≥ 2. Subtract 2: (x−1−2x−4)/(x+2) ≥ 0, so (−x−5)/(x+2) ≥ 0, equivalent to (x+5)/(x+2) ≤ 0. Critical values are −5 and −2. Testing intervals gives positive on (−∞,−5), negative on (−5,−2), positive on (−2,∞). Include x=−5 because the numerator is zero; exclude x=−2. Solution: [−5,−2).
Traps
Multiplying by x+2 without checking its sign reverses or invents solutions. Forgetting to exclude a denominator zero is a standard loss. In radical equations, a candidate that makes the radicand negative must be rejected. Do not write a union when the inequality asks for a single interval.