Review of Algebra

1. The Building Blocks: Variables, Constants, and Expressions

In algebra, a variable stands for an unknown quantity, typically represented by letters such as x, y, or z. A constant is a fixed number, like 3, -5, or . By combining variables, constants, and arithmetic operations (+, , , , exponentiation), we obtain expressions. For example, 3x 4y + 7 is an expression that can be evaluated once the values of x and y are known.

2. Solving Linear Equations

A linear equation is an equation of the first degree; its highest exponent on any variable is 1. The standard form is ax + b = c. Solving means isolating the variable:

• Subtract b from both sides.
• Divide by a.

Because algebraic manipulation follows the same rules on both sides of the equals sign, the solution set is preserved.

3. Inequalities and Their Graphs

Inequalities (<, ≤, >, ≥) describe ranges rather than single values. The solution set of a linear inequality is a halfline on the number line. When multiplying or dividing by a negative number, the inequality sign reverses.

On a coordinate plane, linear inequalities produce shaded regions bounded by a line.

4. Functions: Mapping Inputs to Outputs

A function assigns each element of a domain a single element of a codomain. The notation f(x) = 2x + 3 reads f of x equals 2x plus 3. Functions can be expressed as formulas, tables, or graphs. Key concepts include:

• Domain: All permissible inputs.
• Range: All possible outputs.
• Intercepts: Points where the graph meets the axes.
• Inverse: A function that reverses the original mapping, denoted f(x), when it exists.

5. Polynomials and Factoring

A polynomial is a sum of terms with nonnegative integer exponents. The degree is the highest exponent present. Factoring breaks a polynomial into a product of simpler polynomials, which is useful for solving equations and simplifying expressions.

Common techniques include:

• Factoring out the greatest common factor (GCF).
• Using the difference of squares: a b = (a b)(a + b).
• Applying the quadratic formula for seconddegree polynomials: ax + bx + c = 0.
• Grouping terms to factor trinomials.

6. Quadratic Equations

Quadratics have the form ax + bx + c = 0. They can be solved by factoring, completing the square, or using the quadratic formula:

x = [-b (b 4ac)] / (2a)

The discriminant = b 4ac determines the nature of the roots:

• > 0: Two distinct real roots.
• = 0: One repeated real root.
• < 0: Two complex conjugate roots.

7. Systems of Equations

Systems involve two or more equations with the same variables. Solutions are points that satisfy every equation simultaneously. Main methods are:

• Substitution: Solve one equation for a variable, substitute into the other.
• Elimination (addition): Add or subtract equations to cancel a variable.
• Graphical: Plot each equation; intersection points are solutions.

8. Exponential and Logarithmic Functions

Exponential functions have the form f(x) = ab with base b > 0, b 1. They model growth and decay processes. The inverse of an exponential function is a logarithm: log_b(y) = x b = y.

Key properties:

• log_b(mn) = log_b(m) + log_b(n)
• log_b(m) = nlog_b(m)
• Change of base: log_b(a) = log_k(a) / log_k(b)

9. Rational Expressions and Equations

A rational expression is a fraction whose numerator and denominator are polynomials, e.g., (x 1)/(x + 2). Simplify by factoring and canceling common factors, remembering that the original denominator cannot be zero.

When solving rational equations, first find a common denominator, multiply every term by it, and then solve the resulting polynomial equation. Always check for extraneous solutions that make any original denominator zero.

10. Quick Checklist for an Algebra Review

• Can you manipulate expressions using the distributive, commutative, and associative properties?
• Do you feel comfortable solving linear equations and inequalities?
• Are you able to identify the domain and range of common functions?
• Can you factor quadratics and apply the quadratic formula correctly?
• Do you know how to solve systems of equations with substitution or elimination?
• Are exponential growth, decay, and logarithms clear to you?
• Can you simplify rational expressions and check for extraneous roots?

If you answered yes to most of these questions, you have a solid foundation in algebra. Keep practicing by working through varied problems, and soon the concepts will become second nature.

Further Learning Resources

For additional practice and deeper insight, consider these free resources:

• Khan Academy Algebra
• PurpleMath Algebra Tutorials
• OpenStax Algebra and Trigonometry (PDF)
• Art of Problem Solving Algebra Wiki