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Cumulative Review: Algebra, Graphing, and Geometry

A free Pre-Algebra lesson from the “Cumulative Review and Test Prep” unit, with a worked example and practice problems including step-by-step solutions.

Later cumulative review connects symbolic algebra with graphs, geometry, data, and probability. The goal is to recognize which relationship is being used and solve with a clear sequence of steps.

What you'll learn

Why it matters: Most real problems mix skills — a tip plus a tax, a discount plus a unit price, or an area plus a perimeter. Practicing them together is what test day and everyday math both look like.

Worked example

Problem. For y = 3x - 2, what is y when x = 5?

  1. Substitute x = 5.
  2. 3(5) - 2 = 15 - 2.
  3. y = 13.

Answer: 13

Practice problems

1. Solve 6x - 5 = 31.

Show solution
  1. Equation Review: First identify exactly what the question is asking: Solve 6x - 5 = 31.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Add 5: 6x = 36.
  4. Divide by 6: x = 6.
  5. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

2. Solve -2x > 10. Enter the boundary number.

Show solution
  1. Inequality Review: First identify exactly what the question is asking: Solve -2x > 10. Enter the boundary number.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Divide by -2 and flip the sign.
  4. x < -5, so the boundary number is -5.
  5. Check the result by substituting or estimating: the response should match -5 and make sense in the original problem.

Answer: -5

3. Which point lies on y = 2x + 3?

Choices: (4, 11) · (4, 8) · (11, 4) · (3, 4)

Show solution
  1. Graphing Review: First identify exactly what the question is asking: Which point lies on y = 2x + 3?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. When x = 4, y = 2(4) + 3 = 11.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (4, 11)

4. Which graph matches x >= 2?

Choices: Closed circle at 2, shade right · Open circle at 2, shade right · Closed circle at 2, shade left · Open circle at 2, shade left

Show solution
  1. Inequality Graph Review: First identify exactly what the question is asking: Which graph matches x >= 2?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. >= includes the boundary.
  4. Greater values are to the right.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Closed circle at 2, shade right

5. Reflect (-6, 2) across the x-axis.

Choices: (-6, -2) · (6, 2) · (6, -2) · (2, -6)

Show solution
  1. Coordinate Review: First identify exactly what the question is asking: Reflect (-6, 2) across the x-axis.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Reflecting across the x-axis changes the sign of y.
  4. The x-coordinate stays -6.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (-6, -2)

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