GED Prep: Algebra and Functions (Linear Equations, Inequalities, Patterns, Function Notation)

GED – Algebra and Functions (Linear Equations, Inequalities, Patterns, Function Notation)

GED Algebra & Functions Study Guide: Linear Equations, Inequalities, Patterns, and Function Notation

What This Is

Algebra and functions make up ~30% of the GED Math test—you’ll see linear equations, inequalities, patterns, and function notation in word problems, graphs, and tables. These skills help you model real-world situations (e.g., calculating a phone plan’s total cost, determining how many hours you need to work to afford a car). A typical test question might ask: "A gym charges a $50 sign-up fee plus $20 per month. Write an equation for the total cost (C) after m months, then find the cost after 6 months."

Key Terms & Rules

• Linear Equation: An equation that forms a straight line when graphed; written as y = mx + b (slope-intercept form).

• Example: y = 3x + 2 (slope = 3, y-intercept = 2).

• Slope (m): The rate of change; rise/run or (y₂ – y₁)/(x₂ – x₁).

• Example: If a line passes through (1, 4) and (3, 10), slope = (10 – 4)/(3 – 1) = 3.

• Y-Intercept (b): The point where the line crosses the y-axis (x = 0).

• Example: In y = 2x – 5, the y-intercept is (0, –5).

• Standard Form: Ax + By = C (A, B, C are integers; A ≥ 0).

• Example: 2x + 3y = 6.

• Inequality Symbols:

• > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to).

• Rule: When multiplying/dividing by a negative number, flip the inequality sign.

  
  
  • Example: –2x > 6 → x < –3.

• Function Notation: f(x) means "f is a function of x." Replace x with the input value.

• Example: If f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7.

• Arithmetic Sequence: A pattern where each term increases by a constant difference (d).

• Formula: aₙ = a₁ + (n – 1)d (aₙ = nth term, a₁ = first term).

• Example: 3, 7, 11, 15… (d = 4).

• Domain & Range:

• Domain: All possible x-values (inputs).
• Range: All possible y-values (outputs).

• Example: For f(x) = √x, domain = x ≥ 0; range = y ≥ 0.

• Parallel Lines: Lines with the same slope (m₁ = m₂).

• Perpendicular Lines: Lines with slopes that are negative reciprocals (m₁ × m₂ = –1).

• Example: m₁ = 2, m₂ = –1/2.

• System of Equations: Two or more equations with the same variables. Solutions are the intersection point(s).

• Methods: Substitution, elimination, or graphing.

Step-by-Step / Process Flow

1. Solving Linear Equations (e.g., 3x + 5 = 20)

1. Isolate the variable term: Subtract 5 from both sides → 3x = 15.
2. Solve for x: Divide by 3 → x = 5.
3. Check: Plug x = 5 back into the original equation → 3(5) + 5 = 20 ✓.

2. Graphing a Line from an Equation (e.g., y = –2x + 3)

1. Identify slope (m) and y-intercept (b): m = –2, b = 3.
2. Plot the y-intercept: (0, 3).
3. Use slope to find another point: From (0, 3), go down 2 (rise = –2), right 1 (run = 1) → (1, 1).
4. Draw the line through the points.

3. Solving Inequalities (e.g., –4x + 7 ≤ 15)

1. Isolate the variable term: Subtract 7 → –4x ≤ 8.
2. Divide by –4 (flip the sign!): x ≥ –2.
3. Graph the solution: Closed circle at –2, shade to the right.

4. Evaluating Functions (e.g., f(x) = x² – 3x; find f(–1))

1. Replace x with –1: f(–1) = (–1)² – 3(–1).
2. Simplify: 1 + 3 = 4.

5. Finding the nth Term of a Sequence (e.g., 5, 9, 13, 17…)

1. Identify the pattern: Each term increases by 4 → d = 4.
2. Use the formula: aₙ = a₁ + (n – 1)d → aₙ = 5 + (n – 1)4.
3. Simplify: aₙ = 4n + 1.

Common Mistakes

• Mistake: Forgetting to flip the inequality sign when multiplying/dividing by a negative.

• Correction: Always flip the sign when multiplying/dividing by a negative number.

  
  
  • Why? –2x > 6 becomes x < –3 because dividing by –2 reverses the inequality.

• Mistake: Mixing up slope and y-intercept in y = mx + b.

• Correction: Remember "m" is for mountain (slope) and "b" is for beginning (y-intercept).

• Mistake: Incorrectly solving for x in equations like 2(x + 3) = 10 by only dividing by 2 first.

• Correction: Distribute first → 2x + 6 = 10, then subtract 6 → 2x = 4, then divide by 2 → x = 2.

• Mistake: Graphing inequalities with the wrong shading (e.g., shading above for y < 2x + 1).

• Correction: For < or ≤, shade below the line; for > or ≥, shade above.

• Mistake: Misinterpreting function notation (e.g., f(2) = 5 means the point (2, 5), not (5, 2)).

• Correction: f(x) = y, so f(2) = 5 → (2, 5).

Exam Insights

• Most-Tested Concepts:
• Slope (calculating from two points or a graph).
• Writing equations from word problems (e.g., "A taxi charges $3 plus $1.50 per mile").
• Solving inequalities and graphing solutions.

• Function notation (e.g., f(x) = 3x – 2; find f(4)).

• Tricky Distractors:

• Reverse slope: The GED might give a line with a negative slope but ask for the "rate of increase" (trick: slope is negative, so it’s decreasing).
• Inequality graphs: Watch for dashed vs. solid lines (dashed = < or >; solid = ≤ or ≥).

• Systems of equations: The GED often includes no solution (parallel lines) or infinite solutions (same line) as answer choices.

• Real-World Applications:

• Budgeting: "You earn $15/hour and have $50 saved. Write an equation for total money (M) after h hours."
• Distance/Rate/Time: "A car travels 60 mph. How far does it go in 2.5 hours?" (d = rt → d = 60 × 2.5 = 150 miles).

Quick Check Questions

1. Which equation represents a line with a slope of –3 and a y-intercept of 4?
   a) y = 4x – 3
   b) y = –3x + 4
   c) y = 3x – 4
   d) y = –4x + 3
   Answer: b) y = –3x + 4 (Slope = –3, y-intercept = 4).

2. Solve for x: 5(x – 2) + 3 = 28
   a) x = 3
   b) x = 5
   c) x = 7
   d) x = 9
   Answer: c) x = 7 (Distribute: 5x – 10 + 3 = 28 → 5x – 7 = 28 → 5x = 35 → x = 7).

3. A sequence starts at 10 and increases by 6 each time. What is the 5th term?
   a) 28
   b) 34
   c) 40
   d) 46
   Answer: b) 34 (aₙ = 10 + (5 – 1)6 = 10 + 24 = 34).

Last-Minute Cram Sheet

1. Slope formula: m = (y₂ – y₁)/(x₂ – x₁).
2. Slope-intercept form: y = mx + b (m = slope, b = y-intercept).
3. Standard form: Ax + By = C (A, B, C are integers).
4. Inequality rule: ⚠️ Flip the sign when multiplying/dividing by a negative.
5. Function notation: f(x) = y (e.g., f(2) = 5 → point (2, 5)).
6. Arithmetic sequence: aₙ = a₁ + (n – 1)d.
7. Parallel lines: Same slope (m₁ = m₂).
8. Perpendicular lines: Slopes are negative reciprocals (m₁ × m₂ = –1).
9. Graphing inequalities: Dashed line for < or >; solid for ≤ or ≥.
10. Systems of equations: Solution is the intersection point (or no solution/infinite solutions).