Overview
This study guide covers the core algebraic concepts tested on the GED Mathematical Reasoning exam. Topics include simplifying and evaluating expressions, solving equations and inequalities, understanding functions, and applying algebra to real-world problems. Mastering these skills requires recognizing patterns, applying properties correctly, and translating word problems into mathematical language.
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Expressions & Operations
Summary
Algebraic expressions are mathematical phrases that combine variables, numbers, and operations. Working with expressions requires knowing the rules for combining like terms, applying the distributive property, and substituting values correctly.
Key Concepts
• Evaluating an expression: Replace each variable with its given value, then simplify using the order of operations (PEMDAS/GEMDAS) — Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
• Like terms: Terms sharing the same variable(s) raised to the same power. Only like terms can be added or subtracted.
- ✅ 3x and 5x are like terms
- ❌ 3x and 3x² are NOT like terms
• Combining like terms: Group and simplify like terms separately.
- Example: 3x + 5y − x + 2y = (3x − x) + (5y + 2y) = 2x + 7y
• Distributive property: Multiply the outside factor by each term inside the parentheses.
- Formula: a(b + c) = ab + ac
- Example: 4(3x − 2) = 12x − 8
Worked Example: Evaluating with Negatives
Evaluate 2x² − 3x + 1 when x = −2:
1. Substitute: 2(−2)² − 3(−2) + 1
2. Exponent first: 2(4) − 3(−2) + 1
3. Multiply: 8 + 6 + 1
4. = 15
Key Terms
• Variable — a letter representing an unknown value
• Coefficient — the number multiplied by a variable (in 3x, the coefficient is 3)
• Constant — a fixed number with no variable
• Expression — a mathematical phrase with no equals sign
• Simplify — reduce an expression to its most compact form
⚠️ Watch Out For
• Squaring negative numbers: (−2)² = +4, NOT −4. The negative is inside the parentheses, so it gets squared too.
• Distributing negatives: In 4(3x − 2), make sure to distribute to both terms: 12x − 8, not 12x − 2.
• Combining unlike terms: 2x + 7y cannot be simplified further — x and y are different variables.
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Solving Equations
Summary
Solving an equation means finding the value of the variable that makes the equation true. The core strategy is to perform inverse operations (opposites) to isolate the variable, always keeping the equation balanced by doing the same thing to both sides.
Key Concepts
• Goal: Isolate the variable on one side of the equation.
• Inverse operations: Addition ↔ Subtraction; Multiplication ↔ Division
• Multi-step equations: Distribute first, then combine like terms, then isolate the variable.
• No solution: If solving produces a contradiction like 3 = 7, no value works — the equation has no solution.
• Systems of equations: Two equations with two variables; solved by substitution or elimination.
Step-by-Step Examples
Basic: Solve 3x + 7 = 22
1. Subtract 7: 3x = 15
2. Divide by 3: x = 5
With distribution: Solve 2(x − 4) = 10
1. Distribute: 2x − 8 = 10
2. Add 8: 2x = 18
3. Divide by 2: x = 9
With fractions: Solve x/4 − 3 = 5
1. Add 3: x/4 = 8
2. Multiply by 4: x = 32
System of equations: y = 2x + 1 and y = −x + 7
1. Set equal: 2x + 1 = −x + 7
2. Add x, subtract 1: 3x = 6 → x = 2
3. Substitute: y = 2(2) + 1 = 5
4. Solution: (2, 5)
Writing Equations from Word Problems
| Phrase | Mathematical Meaning |
|---|---|
| "per," "each," "for every" | Multiplication (rate × variable) |
| "total," "altogether" | Addition |
| "fee," "flat charge," "one-time cost" | Constant added to expression |
| "how many" or "find the amount" | Set expression = total and solve |
Example: Car rental costs $25/day plus $40 fee. Total = $165.
• Equation: 25d + 40 = 165 → d = 5 days
Key Terms
• Equation — a mathematical statement with an equals sign showing two expressions are equal
• Solution — the value of the variable that makes the equation true
• Inverse operation — the opposite operation used to undo a step
• System of equations — a set of two or more equations with the same variables
⚠️ Watch Out For
• Forgetting to distribute before moving terms: In 2(x − 4) = 10, distribute before adding 8.
• Plumber/tiered pricing problems: The first hour may cost differently — write the equation as 75 + 50(h − 1) = 275, not 50h = 275.
• Dividing fractions: To solve x/4 = 8, multiply both sides by 4, don't divide.
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Inequalities
Summary
Inequalities express a range of values rather than one exact answer. The solving process mirrors equations with one critical exception: multiplying or dividing by a negative number reverses the inequality sign. Solutions are often shown on a number line.
Key Concepts
• Inequality symbols:
- > greater than (open circle on number line)
- < less than (open circle on number line)
- ≥ greater than or equal to (closed/filled circle)
- ≤ less than or equal to (closed/filled circle)
• Solving process: Same as equations, except flip the sign when multiplying or dividing by a negative.
Step-by-Step Examples
Solve 2x − 3 > 7:
1. Add 3: 2x > 10
2. Divide by 2: x > 5
3. Number line: open circle at 5, shade right
Solve −3x ≤ 12:
1. Divide by −3 (FLIP the sign!): x ≥ −4
2. Number line: closed circle at −4, shade right
Number Line Quick Reference
| Solution | Circle Type | Shading Direction |
|---|---|---|
| x > 5 | Open ○ | Right |
| x < 5 | Open ○ | Left |
| x ≥ 5 | Closed ● | Right |
| x ≤ 5 | Closed ● | Left |
Real-World Meaning
• x ≥ 3 hours worked → must work at least 3 hours
• x ≤ 10 items → can have no more than 10 items
• x > 0 profit → must make more than $0
Key Terms
• Inequality — a statement showing that two expressions are not necessarily equal, using <, >, ≤, or ≥
• Open circle — indicates the boundary point is not included in the solution
• Closed circle — indicates the boundary point is included in the solution
⚠️ Watch Out For
• The flip rule: Only flip the inequality sign when multiplying or dividing by a negative — NOT when adding or subtracting a negative.
• Adding/subtracting negatives: Solving x − 5 > 3 → x > 8. No flip needed because you only added 5.
• Graphing direction: x > 5 shades to the right (larger numbers), x < 5 shades to the left.
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Functions & Patterns
Summary
A function is a special relationship between inputs and outputs where every input has exactly one output. Linear functions are the most common type on the GED, described by the slope-intercept form y = mx + b.
Key Concepts
• Function definition: Each x-value maps to exactly one y-value. One input → one output.
• Vertical line test: If any vertical line hits the graph in more than one point, it is NOT a function.
• Function notation: f(x) means "the output when the input is x."
- f(4) = 3(4) − 5 = 7 means the output is 7 when x = 4.
Slope-Intercept Form: y = mx + b
| Part | Symbol | Meaning |
|---|---|---|
| Slope | m | Rate of change; rise over run |
| y-intercept | b | Value of y when x = 0; starting point |
Slope formula:
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$$
Example: Points (1, 2) and (4, 8):
• Slope = (8 − 2)/(4 − 1) = 6/3 = 2
Interpreting Slope in Context
| Slope Type | What It Means |
|---|---|
| Positive slope | Value is increasing (going up) |
| Negative slope | Value is decreasing (going down) |
| Zero slope | Value is constant (flat line) |
| Steep slope | Fast rate of change |
| Gentle slope | Slow rate of change |
Real-world example: A negative slope on a gas tank graph means the fuel level is decreasing at a constant rate.
Key Terms
• Function — a rule that assigns exactly one output to each input
• Input/Domain — the x-values (what you put in)
• Output/Range — the y-values (what you get out)
• Slope — the steepness and direction of a line; rate of change
• y-intercept — where the line crosses the y-axis (x = 0)
• Linear function — a function whose graph is a straight line
⚠️ Watch Out For
• Vertical line test confusion: A vertical line tests for functions — don't confuse it with slope direction.
• Slope calculation order: Always subtract in the same order: (y₂ − y₁)/(x₂ − x₁). Mixing order gives a wrong sign.
• Negative slopes: A negative slope is still a valid function — "negative" refers to direction, not validity.
• f(x) notation: f(4) does NOT mean f times 4. It means "evaluate the function at x = 4."
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Real-World Applications
Summary
The GED frequently presents algebra through word problems. The key skill is translating words into expressions or equations, solving them, and interpreting the answer in context.
Common Problem Types & Strategies
#### 1. Writing Expressions
Break the problem into parts and identify the variable.
• Maria's pay: $12/hour + $50 bonus → 12h + 50
• Store markup: Original cost c, marked up 30% → c + 0.30c = 1.30c
#### 2. Setting Up and Solving Equations
Look for the condition that makes two quantities equal.
• Ana & Ben savings: 200 + 15w = 80 + 25w → w = 12 weeks
#### 3. Tiered/Stepped Pricing
Identify the flat fee and the per-unit charge carefully.
• Plumber: $75 first hour + $50 each additional hour
- Equation: 75 + 50(h − 1) = 275 → h = 5 hours
#### 4. Proportions and Ratios
Set up equivalent fractions and cross-multiply.
• Recipe: 3:2 flour:sugar ratio, 9 cups flour → 3/2 = 9/x → x = 6 cups sugar
Word Problem Translation Guide
| English Phrase | Algebra |
|---|---|
| "per," "for each," "rate of" | Multiply by variable |
| "total," "sum," "combined" | Addition |
| "difference," "less than" | Subtraction |
| "times as much" | Multiplication |
| "split equally," "each gets" | Division |
| "at least" | ≥ |
| "no more than," "at most" | ≤ |
| "will have the same" | Set expressions equal (=) |
⚠️ Watch Out For
• Tiered pricing: "Each additional hour" means the first hour is separate. Use (h − 1) for additional hours.
• Markup vs. markup price: A 30% markup on cost c gives a selling price of 1.30c, not just 0.30c.
• Proportion setup: Keep ratios consistent — flour/sugar = flour/sugar, not flour/sugar = sugar/flour.
• Units in the answer: Always re-read the question — are they asking for weeks, hours, cups, or dollars?
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Quick Review Checklist
Use this checklist before your exam to confirm you understand each major concept:
• [ ] I can evaluate an expression by substituting values and following order of operations (PEMDAS)
• [ ] I can identify and combine like terms (same variable, same exponent)
• [ ] I can apply the distributive property correctly, including with negatives
• [ ] I can solve multi-step linear equations by using inverse operations on both sides
• [ ] I can set up equations from word problems by identifying the variable and what equals what
• [ ] I can solve a system of equations by substitution (set equal, solve for x, then find y)
• [ ] I know when an equation has no solution (contradiction like 3 = 7)
• [ ] I can solve inequalities and know when to flip the inequality sign (dividing/multiplying by a negative)
• [ ] I can graph inequalities on a number line using open and closed circles correctly
• [ ] I can define a function and apply the vertical line test
• [ ] I can calculate slope using the formula (y₂ − y₁)/(x₂ − x₁)
• [ ] I can identify the slope and y-intercept from y = mx + b and interpret them in context
• [ ] I can evaluate function notation like f(4)
• [ ] I can translate word problems into algebraic expressions and equations
• [ ] I can solve proportion problems using cross-multiplication
• [ ] I can handle tiered pricing problems by carefully separating the flat fee from the per-unit charge
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Study Tip: For each topic, practice writing your own word problem and solving it. If you can create the problem AND solve it, you truly understand the concept.