Patterns and Algebra
What to know
Patterns and algebra covers sequences, functions, and symbolic manipulation. Recognize arithmetic and geometric patterns and write rules for the nth term. Move fluently among tables, graphs, equations, and verbal descriptions of linear relationships; compute slope and intercepts; solve multi-step equations, inequalities, and systems; and apply exponent laws, polynomial operations, factoring, and quadratic relationships. Proportional and nonproportional linear situations are a frequent context, so distinguish additive from multiplicative change.
Traps
Sign errors dominate. Distribute a negative across every term: −2(x−3) = −2x+6, not −2x−6. Reverse the inequality symbol when multiplying or dividing by a negative. Slope is change in y over change in x; a common distractor inverts it. Do not confuse the y-intercept with the x-intercept, or a function's value with its input. Exponent rules apply to products and quotients of powers, not to sums: x² + x² = 2x², not x⁴.
Worked example
Solve 3(x−2) = 2x+5. Distribute: 3x−6 = 2x+5. Subtract 2x: x−6 = 5. Add 6: x = 11. Check: 3(9) = 27 and 2(11)+5 = 27. Rule of thumb: after solving, substitute back-especially when the original equation contains parentheses or fractions.
For slope through (2,3) and (6,11): (11−3)/(6−2) = 8/4 = 2, meaning y increases 2 for every 1 increase in x.