Solving Linear Equations and Inequalities
Key Concepts
- One-variable and multi-variable linear equations
- Inequalities and compound inequalities
- Absolute value equations and inequalities
Skills Tested
- Solving for a variable
- Interpreting and graphing solution sets
- Applying properties of equality and inequality
Example Question
Solve for x: 3(x - 2) + 5 = 2x + 7 A. 1 B. 3 C. Answer: 6 (Distribute and solve: 3x - 6 + 5 = 2x + 7 \Rightarrow x = 6)
Solving Quadratic Equations
Key Concepts
- Factoring, completing the square, and quadratic formula
- Recognizing standard form: ax^2 + bx + c = 0
Skills Tested
- Solving quadratic equations using various methods
- Finding real or complex roots
- Working with discriminants to determine number/type of solutions
Example Question
What are the solutions to x^2 - 5x + 6 = 0? A. x = 3, 2 B. x = -3, -2 C. Answer: A (Factors as (x - 3)(x - 2) = 0)
Expressions, Polynomials, and Rational Algebra
Key Concepts
- Adding, subtracting, multiplying polynomials
- Factoring trinomials and special products (e.g., difference of squares)
- Simplifying rational expressions (fractions with variables)
Skills Tested
- Performing algebraic operations correctly
- Recognizing factoring patterns
- Simplifying complex fractions
Example Question
Simplify: \frac{x^2 - 4}{x - 2} A. x + 2 B. x - 2 C. Answer: x + 2 (Factor numerator: (x - 2)(x + 2), then cancel)
Systems of Equations
Key Concepts
- Solving systems of two linear equations or linear + nonlinear
- Substitution, elimination, or graphical methods
- Recognizing number of solutions (one, none, infinite)
Skills Tested
- Solving 2-variable systems
- Interpreting intersections as solutions
- Identifying consistent/inconsistent systems
Example Question
Solve: y = 2x + 1 y = -x + 4 A. (1, 3) B. (2, 5) C. Answer: (1, 3) (Set equal: 2x + 1 = -x + 4 \Rightarrow x = 1, then y = 3)
Algebraic Formulas and Expressions
Key Concepts
- Evaluating expressions for given values
- Using formulas (e.g., distance, interest, slope)
- Translating word problems into equations
Skills Tested
- Plugging into formulas correctly
- Solving for an unknown in a given formula
- Recognizing correct setup from verbal description
Example Question
If A = P(1 + rt), what is P when A = 660, r = 0.1, t = 2? A. 500 B. 600 C. Answer: 600 (Solve: 660 = P(1 + 0.1 × 2) = P(1.2) \Rightarrow P = 550)
Graphing and Interpreting Linear Equations
Key Concepts
- Slope-intercept form: y = mx + b
- Point-slope and standard forms
- Parallel/perpendicular lines
- Graph interpretation
Skills Tested
- Calculating slope from points or equations
- Graphing linear functions
- Understanding transformations (shifts, reflections)
Example Question
What is the slope of a line perpendicular to y = \frac{2}{3}x + 5? A. -\frac{2}{3} B. \frac{3}{2} C. Answer: -\frac{3}{2} (Negative reciprocal of \frac{2}{3})