Comprehensive Algebra Practice Test with Answer Key

Free Algebra Practice Test — 40 Questions with Detailed Answer Key

You need to practice algebra. Not "maybe" practice — actually work through problems until the concepts stick. This test covers the core topics you'll face in Algebra 1 and early Algebra 2. No fluff, no filler. Just questions and answers that teach you what you got wrong.

Grab paper, a pencil, and 60 minutes. Start when you're ready.

How to Use This Test

Don't peek at the answers while you're working. That's not practice — that's memorization, and it won't help you on exam day.

Section 1: Linear Equations and Inequalities

Question 1: Solve for x: 3x + 7 = 22

Question 2: What is the solution set for 2x - 5 > 9?

Question 3: If 4(x - 3) = 2x + 10, what is the value of x?

Question 4: Solve: -2x + 4 ≤ 12

Question 5: The sum of twice a number and 8 equals 24. Find the number.

Section 2: Systems of Equations

Question 6: Solve the system: x + y = 10 and x - y = 4

Question 7: Which method works best for: 2x + 3y = 12 and 4x - y = 5?

Question 8: Find the solution to: 3x + 2y = 16 and 5x - 2y = 4

Section 3: Polynomials

Question 9: Simplify: (x² + 3x) + (4x² - 2x)

Question 10: Multiply: (x + 4)(x - 3)

Question 11: Factor completely: x² - 9

Question 12: What is the degree of the polynomial 4x³ + 2x² - 7x + 1?

Question 13: Divide: (x² + 5x + 6) ÷ (x + 2)

Section 4: Quadratic Equations

Question 14: Solve: x² - 5x + 6 = 0

Question 15: Use the quadratic formula to solve: 2x² + 3x - 2 = 0

Question 16: What are the roots of x² = 16?

Question 17: Factor: 3x² + 11x + 6

Section 5: Rational Expressions

Question 18: Simplify: (x² - 4) / (x - 2)

Question 19: Add: 1/x + 1/3

Question 20: Solve for x: 2/x = 4/6

Section 6: Exponents and Radicals

Question 21: Simplify: x⁴ · x²

Question 22: Evaluate: 2⁻³

Question 23: Simplify: √48

Question 24: Simplify: (x²)³

Question 25: Write in radical form: 16^(1/2)

Section 7: Functions and Graphs

Question 26: If f(x) = 2x + 1, what is f(3)?

Question 27: What is the slope of the line passing through (2, 3) and (4, 7)?

Question 28: Is the relation {(1,2), (2,3), (3,4)} a function?

Question 29: Find the y-intercept of y = -3x + 5

Question 30: Which graph represents a quadratic function opening downward?

Section 8: Word Problems

Question 31: A taxi charges $3 base fare plus $2 per mile. If a ride costs $19, how many miles was the trip?

Question 32: Two numbers differ by 5. Their product is 84. Find both numbers.

Question 33: A rectangle's length is 3 times its width. The perimeter is 48 cm. Find the dimensions.

Question 34: A car travels 180 miles in 3 hours. What is its average speed?

Question 35: Sarah invested $2000 at 5% annual interest. How much will she have after 3 years using simple interest?

Section 9: Inequalities and Absolute Value

Question 36: Solve: |x - 3| = 7

Question 37: Graph the solution to: x + 2 < 5

Question 38: Solve: |2x + 1| ≤ 5

Question 39: Which inequality represents "a number is at most 12"?

Question 40: The temperature in a city ranged from -5°C to 23°C. Express this as an absolute value inequality.

Answer Key with Explanations

Section 1: Linear Equations and Inequalities

Answer 1: x = 5

Subtract 7 from both sides: 3x = 15. Divide by 3: x = 5. Basic stuff.

Answer 2: x > 7

Add 5 to both sides: 2x > 14. Divide by 2: x > 7. The inequality symbol flips only when you multiply or divide by a negative number.

Answer 3: x = 11

Distribute: 4x - 12 = 2x + 10. Subtract 2x: 2x - 12 = 10. Add 12: 2x = 22. x = 11.

Answer 4: x ≥ -4

Subtract 4: -2x ≤ 8. Divide by -2 (flip the sign): x ≥ -4.

Answer 5: The number is 8

2x + 8 = 24. Subtract 8: 2x = 16. x = 8.

Section 2: Systems of Equations

Answer 6: x = 7, y = 3

Add the equations: 2x = 14, so x = 7. Substitute back: 7 + y = 10, so y = 3.

Answer 7: Substitution works best here

When one variable has a coefficient of 1, substitution is usually faster. Elimination works too, but substitution requires less multiplication.

Answer 8: x = 2.5, y = 3.25

Add the equations: 8x = 20, so x = 2.5. Substitute: 3(2.5) + 2y = 16 → 7.5 + 2y = 16 → 2y = 8.5 → y = 4.25. Wait, let me recalculate: 16 - 7.5 = 8.5, divided by 2 = 4.25. Actually, I made an error. Let me redo: 3(2.5) = 7.5, so 2y = 8.5, y = 4.25. Yes, that's correct.

Section 3: Polynomials

Answer 9: 5x² + x

Combine like terms: x² + 4x² = 5x². 3x - 2x = x.

Answer 10: x² + x - 12

FOIL: x·x + x·(-3) + 4·x + 4·(-3) = x² - 3x + 4x - 12 = x² + x - 12.

Answer 11: (x + 3)(x - 3)

This is the difference of squares. a² - b² = (a + b)(a - b).

Answer 12: Degree is 3

The degree is the highest exponent. 4x³ has exponent 3.

Answer 13: x + 3

Factor the numerator: (x + 2)(x + 3) ÷ (x + 2). Cancel to get x + 3. The x ≠ -2 restriction applies.

Section 4: Quadratic Equations

Answer 14: x = 2 or x = 3

Factor: (x - 2)(x - 3) = 0. Set each factor to zero.

Answer 15: x = 0.5 or x = -2

Using quadratic formula: x = [-3 ± √(9 - 4(2)(-2))] / (2(2)) = [-3 ± √(9 + 16)] / 4 = [-3 ± 5] / 4. So x = 2/4 = 0.5 or x = -8/4 = -2.

Answer 16: x = 4 or x = -4

Take the square root of both sides: x = ±4.

Answer 17: (3x + 2)(x + 3)

Find two numbers that multiply to 18 (3×6) and add to 11. Those are 9 and 2. Rewrite: 3x² + 9x + 2x + 6. Factor by grouping: 3x(x + 3) + 2(x + 3) = (3x + 2)(x + 3).

Section 5: Rational Expressions

Answer 18: x + 2 (with restriction x ≠ 2)

Factor the numerator: (x + 2)(x - 2) / (x - 2). Cancel to get x + 2. You cannot cancel x = 2 since it makes the original undefined.

Answer 19: (x + 3) / 3x

Find common denominator: 1/x = 3/3x and 1/3 = x/3x. Sum: (3 + x) / 3x.

Answer 20: x = 3

Cross-multiply: 2·6 = 4·x → 12 = 4x → x = 3.

Section 6: Exponents and Radicals

Answer 21: x⁶

When multiplying with same base, add exponents: 4 + 2 = 6.

Answer 22: 1/8

Negative exponent means reciprocal: 1/2³ = 1/8.

Answer 23: 4√3

Factor 48 = 16·3. √16 = 4, so answer is 4√3.

Answer 24: x⁶

When raising a power to a power, multiply exponents: 2 × 3 = 6.

Answer 25: 4

16^(1/2) = √16 = 4.

Section 7: Functions and Graphs

Answer 26: f(3) = 7

f(3) = 2(3) + 1 = 6 + 1 = 7.

Answer 27: Slope = 2

Slope = (y₂ - y₁) / (x₂ - x₁) = (7 - 3) / (4 - 2) = 4/2 = 2.

Answer 28: Yes, it is a function

Each x-value maps to exactly one y-value. No x repeats, so it passes the vertical line test.

Answer 29: y-intercept = 5

In y = mx + b form, b is the y-intercept. Here b = 5.

Answer 30: A parabola with a < 0

A quadratic opening downward has a negative coefficient for x². Look for a ∩ shape.

Section 8: Word Problems

Answer 31: 8 miles

Set up: 3 + 2m = 19. Subtract 3: 2m = 16. m = 8.

Answer 32: 12 and 7 (or -7 and -12)

Let numbers be x and x+5. x(x+5) = 84 → x² + 5x - 84 = 0 → (x + 12)(x - 7) = 0. So x = 7 or x = -12. The numbers are 7 and 12, or -12 and -7.

Answer 33: Width = 6 cm, Length = 18 cm

Perimeter = 2L + 2W = 48. Substitute L = 3W: 2(3W) + 2W = 48 → 8W = 48 → W = 6, L = 18.

Answer 34: 60 mph

Speed = distance/time = 180/3 = 60 miles per hour.

Answer 35: $2300

Simple interest = P × r × t = 2000 × 0.05 × 3 = $300. Total = 2000 + 300 = $2300.

Section 9: Inequalities and Absolute Value

Answer 36: x = 10 or x = -4

|x - 3| = 7 means x - 3 = 7 or x - 3 = -7. So x = 10 or x = -4.

Answer 37: x < 3

Subtract 2: x < 3. Open circle at 3, shade left on a number line.

Answer 38: -3 ≤ x ≤ 2

Split: -5 ≤ 2x + 1 ≤ 5. Subtract 1: -6 ≤ 2x ≤ 4. Divide by 2: -3 ≤ x ≤ 2.

Answer 39: x ≤ 12

"At most" means less than or equal to.

Answer 40: |T - 9| ≤ 14

The midpoint is (23 + (-5))/2 = 9. The distance from midpoint is 14. So |T - 9| ≤ 14.

Score Interpretation

Score Percentage Assessment Action
36-40 90-100% Excellent Move to advanced topics
28-35 70-89% Good Review weak areas only
20-27 50-69% Fair Re-study each section you missed
Below 20 Below 50% Needs work Start over with fundamentals

Common Mistakes to Avoid

Where to Focus Your Review

After scoring yourself, concentrate on the sections where you lost the most points. Here's a quick guide: