Patterns, Relations, and Algebra
Patterns, Relations, and Algebra
This domain tests pattern recognition, translating words into expressions, evaluating and simplifying expressions, solving one-variable equations, and interpreting relationships such as tables, graphs, and proportional reasoning.
Core rules
- Arithmetic sequences add a constant difference; geometric sequences multiply by a constant ratio. Find the rule, then test it on at least two consecutive pairs.
- Variables represent numbers, not labels. 3n means 3 × n; n + 3 means three more than n.
- Distributive property: a(b + c) = ab + ac. Combine like terms only when the variable part matches: 4x + 3x = 7x, but 4x + 3 cannot simplify.
- Inverse operations solve equations: undo addition with subtraction, multiplication with division, and keep both sides balanced.
- Proportional relationships have a constant ratio y/x = k; their graphs are straight lines through the origin. Nonproportional linear relationships have a nonzero y-intercept.
- Inequalities: multiplying or dividing both sides by a negative number reverses the inequality sign.
Worked example
Solve 3(x + 4) = 27. Distribute: 3x + 12 = 27. Subtract 12: 3x = 15. Divide by 3: x = 5. Check: 3(5 + 4) = 3(9) = 27. The distributive step is valid because 3 multiplies the whole quantity x + 4, not just x.
Traps
Do not distribute to only the first term inside parentheses. Do not add unlike terms. Do not assume every table is proportional; check whether y/x is constant. When solving, perform the same operation on both sides, and reverse the inequality sign only for multiplication or division by a negative.