Algebraic Thinking
Core concepts
Algebraic thinking on this test means understanding a variable as either an unknown value or a quantity that changes, distinguishing expressions (no equals sign) from equations and inequalities, and using the properties of operations: commutative, associative, distributive, identity, and inverse. You should translate verbal phrases into symbols, evaluate expressions by substitution, solve one- and two-step equations and inequalities, describe patterns and rules in tables, and interpret simple proportional relationships and graphed points.
Common traps
- A coefficient sits next to a variable: 3x means 3 times x. Note that x² is not 2x, and 3x + 4 is not 7x.
- Order matters in subtraction and division phrases. "5 less than n" is n − 5, not 5 − n. "Divided by" reverses the same way.
- Distributing a negative changes every sign inside the parentheses: −2(x − 5) = −2x + 10.
- Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.
- Undo addition and subtraction before multiplication and division when solving two-step equations.
Worked example
Solve 4(x − 2) = 3x + 7. Distribute: 4x − 8 = 3x + 7. Subtract 3x from both sides: x − 8 = 7. Add 8: x = 15. Check by substitution: 4(15 − 2) = 4(13) = 52, and 3(15) + 7 = 52. Both sides match.
For an inequality: −2x > 8 becomes x < −4, because dividing by −2 flips the symbol.
Rule of thumb: perform the same operation on both sides, one step per line, and always substitute your answer back into the original equation to verify.