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College Algebra Readiness and Final Test

A free College Algebra lesson from the “Modeling and Readiness” unit, with a worked example and practice problems including step-by-step solutions.

The College Algebra readiness test samples linear equations, functions, quadratics, complex numbers, radicals, rational expressions, conics, polynomials, transformations, rational exponents, logs, and modeling.

What you'll learn

Why it matters: The final readiness test is the trust check for College Algebra: choose the right tool, interpret the structure, and protect the domain or context. It prepares students for placement, precalculus, and applied math where mixed strategy matters more than isolated recall.

Worked example

Problem. A value starts at 500 and grows by 4% each period. What multiplier is used?

  1. Growth means add the percent to 1.
  2. 4% = 0.04.
  3. The multiplier is 1.04.

Answer: 1.04

Practice problems

1. Solve 5x - 4 = 31.

Show solution
  1. Course Review: First identify exactly what the question is asking: Solve 5x - 4 = 31.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Add 4 and divide by 5.
  4. Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.

Answer: 7

2. Find the slope through (1, 2) and (5, 14).

Show solution
  1. Course Review: First identify exactly what the question is asking: Find the slope through (1, 2) and (5, 14).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. 12/4 = 3.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

3. If f(x) = x^2 - 1, find f(8).

Show solution
  1. Course Review: First identify exactly what the question is asking: If f(x) = x^2 - 1, find f(8).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. 64 - 1 = 63.
  4. Check the result by substituting or estimating: the response should match 63 and make sense in the original problem.

Answer: 63

4. Factor x^2 - 5x - 14.

Choices: (x - 7)(x + 2) · (x + 7)(x - 2) · (x - 14)(x + 1) · (x - 5)(x + 14)

Show solution
  1. Course Review: First identify exactly what the question is asking: Factor x^2 - 5x - 14.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. -7 and 2 multiply to -14 and add to -5.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (x - 7)(x + 2)

5. Simplify sqrt(98).

Show solution
  1. Course Review: First identify exactly what the question is asking: Simplify sqrt(98).
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. 98 = 49 x 2.
  4. Check the result by substituting or estimating: the response should match 7sqrt(2) and make sense in the original problem.

Answer: 7sqrt(2)

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