Algebra
Core ideas
Algebra questions focus on linear equations and inequalities, systems of equations, polynomial operations, factoring, and quadratic equations. For a linear equation, isolate the variable by applying inverse operations in reverse order. For inequalities, the solution set reverses direction when you multiply or divide both sides by a negative number. A system of two linear equations can have one solution, no solution (parallel lines with different intercepts), or infinitely many solutions (the same line). With polynomials, combine like terms and remember the distributive property; factoring reverses expansion. Quadratics can be solved by factoring, completing the square, or the quadratic formula x = [−b ± √(b^2−4ac)]/(2a).
Traps
Watch signs when distributing a negative: −2(x−3) = −2x+6, not −2x−6. When solving a system by substitution, substitute the entire expression, not just its first term. Check whether a proposed solution makes a denominator zero or violates the original inequality direction. In the quadratic formula, −b means the opposite of b, and the entire numerator is divided by 2a. Factoring x^2+bx+c requires two numbers whose product is c and sum is b, so sign pairs matter.
Worked example
Solve 3(x−2)+4 = 2x+5. Distribute: 3x−6+4 = 2x+5, so 3x−2 = 2x+5. Subtract 2x from both sides: x−2 = 5. Add 2: x = 7. Check: 3(5)+4 = 19 and 2(7)+5 = 19. Rule of thumb: simplify each side completely, collect variables on one side, then check by substitution.