Algebraic Thinking and Applications
Expect heavy coverage of manipulating expressions and solving equations and inequalities. Key concepts: polynomial operations and factoring (GCF, difference of squares, trinomials, grouping), rational expressions and their domain restrictions, radical equations, and systems of equations. You should also model word problems-rate, mixture, work, and distance problems-by translating relationships into equations.
Traps to watch
- Extraneous solutions: squaring both sides of a radical equation or multiplying by a variable expression can introduce solutions that fail the original. Always check.
- Inequality sign flips: multiplying or dividing by a negative reverses the inequality. Also remember "and" vs. "or" for compound inequalities.
- Domain exclusions: a rational expression is undefined where its denominator is zero, even after simplification.
- Factoring errors: $a^2+b^2$ does not factor over the reals; only the difference of squares does.
Worked example / rule of thumb
Solve $\sqrt{2x+3} = x$. Square: $2x+3 = x^2 \Rightarrow x^2 - 2x - 3 = 0 \Rightarrow (x-3)(x+1)=0$, so $x=3$ or $x=-1$. Check: $x=3$ gives $\sqrt{9}=3$ ✓; $x=-1$ gives $\sqrt{1}=1 \neq -1$ ✗. Only $x=3$. Rule of thumb: after any operation that is not reversible (squaring, clearing denominators), substitute back into the original equation. For word problems, define the variable explicitly and check that the answer is physically sensible (no negative lengths or times).