Overview
This study guide covers the core algebraic concepts tested on the Praxis Core Math exam, including simplifying and evaluating expressions, solving equations and inequalities, understanding functions, graphing linear relationships, and recognizing patterns. Mastery of these topics requires both procedural fluency (knowing how to solve) and conceptual understanding (knowing why methods work). Use this guide alongside practice problems for best results.
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Expressions & Operations
Summary
Algebraic expressions are mathematical phrases combining variables, coefficients, and operations. Simplifying expressions requires recognizing like terms, applying exponent rules, and using the distributive property. Factoring reverses expansion and is critical for solving equations.
Key Concepts
#### Combining Like Terms
• Like terms share the same variable(s) raised to the same power
• Only coefficients are added or subtracted; the variable part does not change
• Example: 3x + 5x − 2x = 6x
#### Distributive Property
• Multiply the term outside by each term inside the parentheses
• Formula: a(b + c) = ab + ac
• Example: 4(3x − 7) = 12x − 28
#### Factoring
• Rewriting an expression as a product of its factors
• Example: x² + 5x + 6 = (x + 2)(x + 3)
• Finding the GCF first simplifies the process
- GCF of 12x²y and 8xy² = 4xy (GCF of coefficients × lowest power of each shared variable)
#### FOIL Method (Multiplying Binomials)
• First, Outer, Inner, Last
• Example: (x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15
#### Exponent Rules
| Rule | Formula | Example |
|------|---------|---------|
| Multiplying same base | xᵃ · xᵇ = xᵃ⁺ᵇ | x³ · x⁴ = x⁷ |
| Dividing same base | xᵃ ÷ xᵇ = xᵃ⁻ᵇ | x⁵ ÷ x² = x³ |
| Power to a power | (xᵃ)ᵇ = xᵃᵇ | (x²)³ = x⁶ |
#### Evaluating Expressions
• Substitute given values for variables, then follow order of operations (PEMDAS)
• Example: 2a² − 3b when a = 3, b = 4 → 2(9) − 12 = 6
Key Terms
• Coefficient – the numerical factor of a term (e.g., 7 in 7x³)
• Variable – a letter representing an unknown value
• Like terms – terms with identical variable parts
• GCF (Greatest Common Factor) – the largest factor shared by two or more terms
• Factoring – rewriting as a product of factors
• FOIL – method for multiplying two binomials
Watch Out For
> ⚠️ Only combine like terms. You cannot add 3x and 3x² — they have different exponents and are NOT like terms.
> ⚠️ Distribute to ALL terms. A common error: 4(3x − 7) ≠ 12x − 7. The 4 must multiply both 3x AND −7.
> ⚠️ Don't forget the negative sign when factoring or distributing. Errors with negatives are among the most frequent mistakes on the exam.
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Solving Equations & Inequalities
Summary
Solving equations means isolating the variable by performing inverse operations in the correct order. Inequalities follow the same rules with one critical exception: the direction of the inequality symbol reverses when multiplying or dividing by a negative number.
Key Concepts
#### Two-Step Equations
• Work backwards through order of operations: undo addition/subtraction first, then multiplication/division
• Example: 3x − 4 = 11 → add 4 → 3x = 15 → divide by 3 → x = 5
• Example: 2x + 9 = 25 → subtract 9 → 2x = 16 → divide by 2 → x = 8
#### Verifying Solutions
• A solution is any value that makes the equation a true statement when substituted
• Always check your answer by plugging it back into the original equation
#### Solving Inequalities
• Follow same steps as equations
• Critical rule: Reverse the inequality symbol when multiplying or dividing by a negative number
• Example: −3x > 12 → divide by −3 → x < −4 (symbol flips!)
#### Quadratic Equations
• Factoring method: Set equation = 0, factor, set each factor = 0
- Example: x² − 9 = 0 → (x + 3)(x − 3) = 0 → x = 3 or x = −3
• Quadratic Formula: Works for any quadratic ax² + bx + c = 0
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
#### Systems of Equations
• Substitution method: Replace one variable using one equation, then solve
• Example: y = 2x + 1 and y = x + 4 → 2x + 1 = x + 4 → x = 3, y = 7
Key Terms
• Inverse operations – operations that undo each other (addition/subtraction, multiplication/division)
• Solution – a value that makes an equation or inequality true
• Inequality – a mathematical statement using <, >, ≤, or ≥
• System of equations – two or more equations solved simultaneously
• Quadratic equation – an equation containing an x² term
• Discriminant – the expression b² − 4ac inside the quadratic formula
Watch Out For
> ⚠️ The inequality flip rule is tested frequently. If you divide or multiply by a negative number, the symbol MUST reverse. Forgetting this is one of the most common exam errors.
> ⚠️ When solving quadratics by factoring, ALWAYS set the equation equal to zero first before factoring.
> ⚠️ When solving systems, substitute back into the original equation to find the second variable — don't assume.
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Functions & Their Properties
Summary
A function is a special type of relationship where every input has exactly one output. Understanding function notation, domain, range, and the difference between function types (linear vs. quadratic) is essential for this section.
Key Concepts
#### Definition of a Function
• Each x-value (input) is paired with exactly one y-value (output)
• A relation is NOT a function if any input maps to more than one output
#### Vertical Line Test
• Draw (or imagine) vertical lines across a graph
• If any vertical line crosses the graph more than once → NOT a function
• If every vertical line crosses at most once → IS a function
#### Function Notation
• f(x) is read "f of x" — it represents the output value for a given input x
• f(x) is interchangeable with y
• Example: If f(x) = 3x − 2, then f(4) = 3(4) − 2 = 10
#### Domain & Range
| Term | Definition | Plain Language |
|------|-----------|---------------|
| Domain | All possible input values (x-values) | "What can I plug in?" |
| Range | All possible output values (y-values) | "What can I get out?" |
#### Types of Functions
| Type | Form | Graph |
|------|------|-------|
| Linear | f(x) = mx + b | Straight line |
| Quadratic | f(x) = ax² + bx + c | Parabola (U-shape) |
Key Terms
• Function – a relation with exactly one output per input
• Domain – the set of all valid input values
• Range – the set of all possible output values
• f(x) notation – function notation representing output at input x
• Vertical line test – graphical test to determine if a relation is a function
• Linear function – produces a straight-line graph
• Quadratic function – contains x²; produces a parabolic graph
Watch Out For
> ⚠️ f(x) does NOT mean f times x. It is a notation meaning "the value of function f at input x."
> ⚠️ Domain restrictions: For the Praxis, watch for situations where a denominator could equal zero or a square root of a negative number would occur — these values are excluded from the domain.
> ⚠️ Not all relations are functions. A circle fails the vertical line test and is NOT a function.
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Linear Relationships & Graphs
Summary
Linear relationships form straight lines on a graph. Understanding slope, intercepts, and the slope-intercept form allows you to graph lines, interpret their behavior, and write equations from given information.
Key Concepts
#### Slope
• Measures the steepness and direction of a line
• Formula: m = (y₂ − y₁) / (x₂ − x₁) = rise/run
• Interpretation of slope:
| Slope Value | Meaning |
|-------------|---------|
| Positive | Line rises left to right |
| Negative | Line falls left to right |
| Zero (m = 0) | Horizontal line |
| Undefined | Vertical line |
#### Slope-Intercept Form
• y = mx + b
- m = slope
- b = y-intercept (where the line crosses the y-axis)
• Easiest form for graphing: start at b, use slope (rise/run) to plot additional points
#### Intercepts
| Intercept | Definition | How to Find |
|-----------|-----------|-------------|
| y-intercept | Where line crosses y-axis | Set x = 0, solve for y |
| x-intercept | Where line crosses x-axis | Set y = 0, solve for x |
#### Special Lines
• Horizontal line: y = constant (e.g., y = 3); slope = 0
• Vertical line: x = constant (e.g., x = 2); slope = undefined
Key Terms
• Slope (m) – rate of change; rise over run
• y-intercept (b) – point where the line meets the y-axis
• x-intercept – point where the line meets the x-axis
• Slope-intercept form – y = mx + b
• Rise – vertical change (Δy)
• Run – horizontal change (Δx)
Watch Out For
> ⚠️ Slope = 0 vs. Undefined slope are different! Horizontal lines have zero slope; vertical lines have undefined slope. These are commonly confused on tests.
> ⚠️ Be careful with slope formula order. Always subtract in the same order: (y₂ − y₁) / (x₂ − x₁). Mixing up the order gives you the wrong sign.
> ⚠️ A steeper line does NOT always mean a larger number if slopes are negative — compare absolute values carefully.
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Patterns & Algebraic Reasoning
Summary
Recognizing patterns is a foundational algebraic skill. Arithmetic sequences follow an additive pattern, while proportions express equal ratios and are solved using cross-multiplication.
Key Concepts
#### Arithmetic Sequences
• Each term is found by adding a constant (common difference) to the previous term
• Example: 4, 9, 14, 19, … → common difference = +5
• To find the nth term: aₙ = a₁ + (n − 1)d
- a₁ = first term, d = common difference, n = term number
#### Identifying Common Difference
• Subtract any term from the term that follows it
• Example: 14 − 9 = 5; 9 − 4 = 5 → common difference = 5
#### Proportions
• A proportion states two ratios are equal: a/b = c/d
• Cross-multiply to solve: a · d = b · c
• Example: 3/x = 6/10 → 3(10) = 6x → 30 = 6x → x = 5
• Applications: unit rates, scaling, percent problems
Key Terms
• Arithmetic sequence – a sequence with a constant difference between terms
• Common difference (d) – the fixed amount added between terms in an arithmetic sequence
• Proportion – an equation stating two ratios are equal
• Cross-multiplication – solving a proportion by multiplying diagonally (a · d = b · c)
• Ratio – a comparison of two quantities
Watch Out For
> ⚠️ Arithmetic vs. Geometric sequences: Arithmetic sequences ADD a constant; geometric sequences MULTIPLY by a constant. Make sure you identify which type you're working with.
> ⚠️ In proportions, units must be consistent. If comparing miles to hours, both ratios must be set up in the same order (miles/hours = miles/hours).
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Quick Review Checklist
Use this checklist before your exam to confirm mastery of each topic:
Expressions & Operations
• [ ] I can combine like terms correctly, including with negative coefficients
• [ ] I can apply the distributive property, including distributing negatives
• [ ] I can multiply binomials using the FOIL method
• [ ] I can find the GCF of terms with variables
• [ ] I can apply the rule for multiplying same-base exponents (add the exponents)
• [ ] I can evaluate expressions by substituting values and following PEMDAS
Solving Equations & Inequalities
• [ ] I can solve two-step linear equations
• [ ] I can verify a solution by substituting back into the equation
• [ ] I know to flip the inequality symbol when multiplying/dividing by a negative
• [ ] I can solve quadratic equations by factoring (difference of squares and trinomials)
• [ ] I can apply the quadratic formula: x = (−b ± √(b² − 4ac)) / (2a)
• [ ] I can solve a system of equations using substitution
Functions & Their Properties
• [ ] I can define a function and explain the "exactly one output" rule
• [ ] I can apply the vertical line test to a graph
• [ ] I understand f(x) notation and can evaluate functions at given inputs
• [ ] I can identify and distinguish domain and range
• [ ] I can distinguish between linear and quadratic functions by their equations and graphs
Linear Relationships & Graphs
• [ ] I can calculate slope from two points using (y₂ − y₁) / (x₂ − x₁)
• [ ] I know slope = 0 for horizontal lines and undefined for vertical lines
• [ ] I can write and interpret equations in slope-intercept form (y = mx + b)
• [ ] I can find both x- and y-intercepts algebraically
Patterns & Algebraic Reasoning
• [ ] I can identify the common difference in an arithmetic sequence
• [ ] I can find missing terms or extend arithmetic sequences
• [ ] I can set up and solve proportions using cross-multiplication
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Focus extra attention on any unchecked items before test day. Good luck on the Praxis Core Math exam!