College Algebra Cumulative Review

Integrate algebra skills through mixed practice, diagnostics, and strategy.

Learning objectives

After this review you should be able to identify methods for mixed problems, move among representations, solve equations and inequalities, analyze functions, build models, check answers, use SymPy strategically, and create a personal study plan.

Motivation: mixed problems are different

In a chapter exercise, the title often tells you the method. Real problems do not. Ask:

  1. What kind of object is this?
  2. What restrictions matter?
  3. Which method might work?
  4. How can I check the answer?

Algebra decision map

If you see… Think about… Possible tools
denominator with variable restrictions and rational functions factor, cancel, check forbidden values
square root domain and extraneous solutions isolate, square, check
variable in exponent exponentials and logarithms rewrite bases, take logs
two equations simultaneous constraints substitution, elimination, graph
graph shifts transformations a*f(b(x-h))+k
word problem modeling cycle variables, units, assumptions

Diagnostic warm-up

  1. Expand (x + 2)(x - 5).
  2. Solve -2 <= 3x + 1 < 10.
  3. Find the domain of sqrt(x - 4)/(x + 1).
  4. Describe g(x)=2(x-3)^2+1 from f(x)=x^2.
  5. Find the line through (0, 4) and (3, 10).
  6. Find the roots of (x - 2)^2(x + 1).
  7. Simplify (x^2 - 9)/(x - 3) and state the warning.
  8. Solve 2^x = 12 exactly.
  9. Solve x + y = 7, x - y = 1.
  10. Choose a model family for repeated 8% yearly growth.
  1. x^2 - 3x - 10.
  2. -1 <= x < 3.
  3. x >= 4.
  4. Vertical stretch by 2, right 3, up 1.
  5. y = 2x + 4.
  6. Roots: 2 and -1; 2 has multiplicity 2.
  7. x + 3, but original excludes x = 3.
  8. x = log(12)/log(2).
  9. (4, 3).
  10. Exponential.

Method-selection examples

Rational equation:

2/(x - 1) = 3/(x + 2)

First list restrictions x != 1, -2, then cross multiply: 2(x+2)=3(x-1), so x=7.

Exponential model:

500(0.85)^t = 200

Use logarithms: t = log(0.4)/log(0.85).

System from words: 18 nickels and dimes worth $1.35 gives:

n + d = 18
5n + 10d = 135

Mixed worked examples

  • Hidden restriction: (x^2 - 16)/(x - 4) simplifies to x + 4, but excludes x = 4.
  • Absolute value: |x - 3| <= 5 gives -2 <= x <= 8.
  • Inverse: if f(x)=2x+5, then f^-1(x)=(x-5)/2.
  • Quadratic vertex: h(t)=-4(t-3)^2+20 has maximum 20 at t=3.
  • Polynomial: (x - 2)^2(x + 3) touches at 2 and crosses at -3.
  • Radical check: sqrt(x+6)=x gives candidates 3,-2; only 3 works.
  • Log domain: log(x-4)=2 requires x>4, so x=e^2+4 is valid.

Error analysis workshop

  1. A student cancels x in (x + 2)/(x + 3) and gets 2/3.
  2. From -2x < 8, a student writes x < -4.
  3. A student writes log(x + 5) = log(x) + log(5).
  4. A student solves sqrt(x + 5)=x-1, gets x=4,-1, and keeps both.
  1. You cannot cancel terms across addition.
  2. Dividing by a negative reverses the inequality: x > -4.
  3. Logarithms do not distribute over addition.
  4. x = -1 is extraneous.

Strategy sheet

  • Equations: undo operations and check.
  • Inequalities: reverse direction when multiplying or dividing by a negative.
  • Domains: watch denominators, even roots, and logarithms.
  • Transformations: read a*f(b(x-h))+k carefully.
  • Polynomials: factors reveal roots and multiplicity.
  • Systems: solutions satisfy every equation.
  • Modeling: define variables and units before solving.
  • SymPy: verify, do not skip reasoning.

Practice sets

  1. Factor x^2 - 9.
  2. Solve 5x + 2 = 22.
  3. Find the slope between (1, 4) and (3, 10).
  4. Solve |2x - 1| < 7.
  5. Find the domain of (x + 2)/(x^2 - 1).
  6. Solve 3^x = 25 exactly.
  7. Solve y = 2x + 1, y = -x + 7.
  8. A taxi trip costs $3.50 plus $2.20 per mile. A ride costs $25.50. How many miles was it?
  1. (x - 3)(x + 3).
  2. x = 4.
  3. Slope 3.
  4. -3 < x < 4.
  5. Exclude x = -1 and x = 1.
  6. x = log(25)/log(3).
  7. (2, 5).
  8. 10 miles.

Personal study plan

If you missed… Review…
domain restrictions inequalities, rational/radical functions, logs
graph features functions, transformations, polynomials
model setup algebraic modeling and systems
exponentials/logs exponentials and logarithms
tool use symbolic computation with SymPy

Browser symbolic practice

Use math.js, Nerdamer, Algebrite, or PyScript to verify answers. Then write what the tool confirmed and what you still had to decide.

Subtopic guided practice and checkpoints

Mixed review is about choosing a method before calculating. Use these labs to practice recognizing problem types.

Lab 1: choose the method

Guess first. Which method fits each problem: factor, isolate, graph, substitute, or use logarithms?

A. 3x - 5 = 16
B. x^2 - 9x + 20 = 0
C. 2^x = 13
D. y = x + 1 and y = -x + 5

Guided exercise.

A isolate the variable
B factor the quadratic
C use logarithms or numerical solving
D substitute or graph the two lines

Checkpoint. For each problem in your current homework, write the method name before the first algebraic step.

Lab 2: execute step by step

Guess first. In a mixed problem set, why is skipping steps risky?

Guided exercise. Solve 2(x + 4) - 3 = 13.

2(x + 4) - 3 = 13
2x + 8 - 3 = 13
2x + 5 = 13
2x = 8
x = 4

Every line changes one thing, so errors are easier to find.

Checkpoint. Solve 5 - 3(2x - 1) = 20, writing one operation per line.

Lab 3: check and repair

Guess first. If your answer does not check, should you erase everything or search for the first wrong line?

Guided exercise.

Use this repair routine:

  1. Substitute the answer into the original problem.
  2. If it fails, compare each line with the previous line.
  3. Look first for sign errors, distribution errors, and illegal cancellation.
  4. Correct the earliest wrong line, then continue.

Checkpoint. Create a deliberate sign error while solving a linear equation. Then use the repair routine to find and fix it.

Lab 4: explain the answer

Guess first. What is missing from an answer that says only x = 3?

Guided exercise.

A complete answer often includes:

solution value or set
restriction check
meaning in context
units, if the problem has units

Checkpoint. Write a complete final sentence for a ticket-sales problem where a = 20 adult tickets and s = 30 student tickets.

Guessing game checkpoint

Using this lesson with edumath and SymPy

Use edumath’s parser and solver as a checking partner after you have selected a method and tried the problem by hand.

from edumath.core import parse_equation
from edumath.solvers import solve_equation_steps

review_solution = solve_equation_steps(parse_equation("2(x + 4) - 3 = 13"))
print(review_solution.render_text())
Answer: x = 4

Method: linear equation

Steps:

1. Original equation:
Eq(2*(x + 4) - 1*3, 13)

2. Expand both sides:
Eq(2*x + 5, 13)

3. Move everything to one side:
Eq(2*x - 8, 0)
This writes the equation in the form ax + b = 0.

4. Move the constant term:
Eq(2*x, 8)

5. Divide by the coefficient of the variable:
Eq(x, 4)

Check:

x = 4: valid
import sympy as sp

x, y = sp.symbols("x y")
checks = {
    "factor": sp.factor(x**2 - 5*x + 6),
    "solve quadratic": sp.solve(sp.Eq(x**2 - 5*x + 6, 0), x),
    "solve system": sp.solve((sp.Eq(x + y, 5), sp.Eq(x - y, 1)), (x, y)),
}
checks
{'factor': (x - 3)*(x - 2),
 'solve quadratic': [2, 3],
 'solve system': {x: 3, y: 2}}
expr = (x**2 - 1)/(x - 1)
sp.cancel(expr), sp.solve(sp.Eq(sp.denom(expr), 0), x)
(x + 1, [1])
from edumath.algebra import expand_question, factor_question

questions = [expand_question((x + 2)*(x - 3)), factor_question(x**2 - 5*x + 6)]
[question.prompt for question in questions]
['Expand `(x - 3)*(x + 2)`.', 'Factor `x**2 - 5*x + 6`.']

Further reading