Algebra is a key component of the ASVAB Mathematics Knowledge Subtest, testing your ability to solve equations, manipulate expressions, and apply mathematical reasoning. This guide covers essential algebra topics with definitions, formulas, and examples to help you succeed.
Algebra Concepts and Terminology
Key Algebra Terms
- Term – A number, variable, or product of numbers and variables. Example: 3x², 5x, and -7.
- Algebraic Expression – A mathematical phrase that includes numbers, variables, and operations but no equal sign. Example: 3x + 5.
- Equation – A mathematical statement with an equal sign that shows two expressions are equal. Example: 2x + 3 = 7.
- Variable – A symbol (usually a letter) that represents an unknown value. Example: x, y, z.
- Coefficient – The number multiplied by a variable. Example: In 4x, 4 is the coefficient.
- Constant – A number in an expression without a variable. Example: In 2x + 5, the constant is 5.
- Polynomial – An algebraic expression made of one or more terms. Examples:
- Monomial: 4x
- Binomial: x² + 3x
- Trinomial: x² – 5x + 6
- Like Terms – Terms with the same variable and exponent. Example: 3x² and -5x² are like terms.
Solving for x (Unknown Variables)
Solving for x means isolating the variable on one side of the equation.
Example: Solve 2x + 3 = 7
- Subtract 3 from both sides: 2x = 4
- Divide both sides by 2: x = 2
Translating Word Problems into Equations
Example: “A number increased by 5 is 12.”
- Equation: x + 5 = 12
- Solve: x = 12 – 5 = 7
Balancing Algebraic Equations
Equations must stay balanced. What you do to one side, you must do to the other.
Example: Solve 3x – 7 = 8
- Add 7 to both sides: 3x = 15
- Divide by 3: x = 5
Solving One-Step and Multi-Step Equations
One-step equations use just one inverse operation, while multi-step equations require multiple operations.
One-Step Example: Solve x + 9 = 14
- Subtract 9: x = 5
Multi-Step Example: Solve 4x – 2 = 10
- Add 2 to both sides: 4x = 12
- Divide by 4: x = 3
Handling Equations with x on Both Sides
Example: Solve 5x + 4 = 3x + 10
- Subtract 3x from both sides: 2x + 4 = 10
- Subtract 4: 2x = 6
- Divide by 2: x = 3
Simplifying Algebraic Expressions
To simplify expressions:
- Use the distributive property: a(b + c) = ab + ac
- Combine like terms: 3x + 5x = 8x
- Apply exponent rules
Example: Simplify 5(x + 2) + 3x
- Expand: 5x + 10 + 3x
- Combine like terms: 8x + 10
Multiplying Binomials (FOIL Method)
FOIL (First, Outer, Inner, Last) expands binomials.
Example: Expand (x + 3)(x – 2)
- Multiply: x² – 2x + 3x – 6
- Simplify: x² + x – 6
Solving Two-Variable Equations
Use substitution or elimination to solve systems of equations.
Example (Substitution Method): Solve y = 2x + 3 and x + y = 7
- Substitute y = 2x + 3 into the second equation: x + (2x + 3) = 7
- Solve for x: 3x + 3 = 7 → 3x = 4 → x = 4/3
Exponents in Algebra
Exponent Rules:
- Multiplication: x^a × x^b = x^(a+b)
- Division: x^a ÷ x^b = x^(a-b)
- Power Rule: (x^a)^b = x^(a × b)
- Negative Exponents: x^(-a) = 1/x^a
Factoring in Algebra
Factoring means writing expressions as a product of terms.
Greatest Common Factor (GCF)
Factor out the largest term that divides all terms evenly.
Example: Factor 6x² + 9x
- Factor out the GCF: 3x(2x + 3)
Factoring Quadratic Equations
Example: Factor x² + 5x + 6
- (x + 2)(x + 3)
Understanding and Solving Inequalities
Inequality Symbols:
- ≠ (not equal to)
-
(greater than)
- < (less than)
- ≥ (greater than or equal to)
- ≤ (less than or equal to)
Example: Solve 5x – 3 ≤ 12
- Add 3: 5x ≤ 15
- Divide by 5: x ≤ 3
Special Rule:
Flip the inequality sign when multiplying or dividing by a negative number.
Final Review
- Understand algebraic terms and operations.
- Balance equations and solve for variables.
- Factor expressions and solve quadratic equations.
- Work with inequalities carefully, remembering sign changes.