Chapter 2. Algebra: Expressions, Equations and Inequalities

Variables and expressions. A variable stands for an unknown or changing number. Simplify by combining like terms and using the distributive law: a(b + c) = ab + ac [2][3].

Solving linear equations. Perform the same operation on both sides to isolate the variable. Example: 3x − 7 = 11 gives 3x = 18, so x = 6.

Inequalities. Solve like equations, but reverse the inequality sign when multiplying or dividing by a negative number. Example: −2x < 8 gives x > −4.

Systems of linear equations. Solve by substitution or elimination. Example: x + y = 5 and x − y = 1 add to give 2x = 6, so x = 3 and y = 2.

Special products and factoring

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

a² − b² = (a − b)(a + b)

Quadratic equations. For ax² + bx + c = 0 with a ≠ 0, the solutions are

x = (−b ± √(b² − 4ac)) / 2a.

The discriminant D = b² − 4ac tells how many real solutions exist: two if D > 0, one repeated if D = 0, none if D < 0 [3].

Example: x² − 5x + 6 = 0 factors as (x − 2)(x − 3) = 0, so x = 2 or x = 3.

Review: (a) Solve 5x + 2 = 3x + 12. (Answer: x = 5.) (b) How many real solutions does x² + 4x + 5 = 0 have? (Answer: none, since D = −4.)