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Quizzes > High School Quizzes > Mathematics

Ace Your Intermediate Algebra Practice Test

Test Your Skills With Detailed Algebra Problems

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting the dynamic Algebra Ace Challenge for high school students.

Solve for x: 2x + 3 = 11.
5
4
2
3
Subtracting 3 from both sides gives 2x = 8, and then dividing by 2 yields x = 4. This step-by-step isolation of the variable confirms the correct answer.
Solve for x: x - 5 = 0.
5
0
-5
1
Adding 5 to both sides isolates the variable, yielding x = 5. This basic linear equation reinforces the concept of inverse operations.
Simplify the expression: 3x + 2x.
x
5x
6x
3x
Combining like terms 3x and 2x gives 5x. This straightforward process demonstrates the importance of adding coefficients when variables are the same.
What is 3²?
6
9
3
8
3² means 3 multiplied by itself, which equals 9. This question reinforces the basic concept of exponentiation.
Find the x-intercept of the line y = 2x - 4.
-4
-2
4
2
Setting y = 0 in the equation 2x - 4 = 0 yields 2x = 4 and therefore x = 2. The x-intercept is where the graph crosses the x-axis, confirming the answer.
Solve for x: 3(x - 2) = 15.
6
8
7
5
Distribute 3 to get 3x - 6 = 15. Adding 6 to both sides gives 3x = 21, and dividing by 3 yields x = 7. This problem tests basic distribution and inverse operations.
Solve the equation: 4x + 5 = 2x + 13.
5
-4
8
4
Subtract 2x from both sides to obtain 2x + 5 = 13, then subtract 5 to get 2x = 8, and finally divide by 2 to find x = 4. This systematic approach underscores the balance in equations.
Factor the quadratic: x² + 5x + 6.
(x - 2)(x - 3)
(x + 2)²
(x + 1)(x + 6)
(x + 2)(x + 3)
The quadratic factors into (x + 2)(x + 3) because 2 and 3 add to 5 and multiply to 6. This method requires finding two numbers that satisfy both the sum and product conditions.
Simplify the expression: 2(x + 3) - 4(x - 1).
6x + 2
-2x - 10
-2x + 10
2x + 10
Expanding yields 2x + 6 and -4x + 4; combining like terms gives -2x + 10. The process highlights distribution and the importance of combining similar terms accurately.
Solve the inequality: 2x - 5 > 3.
x ≤ 4
x ≥ 4
x < 4
x > 4
Adding 5 to both sides produces 2x > 8, and dividing by 2 leads to x > 4. The solution is reached by performing equivalent operations on both sides without reversing the inequality sign.
If f(x) = 2x + 1, what is the value of f(3)?
8
6
7
5
Substituting 3 into the function gives f(3) = 2(3) + 1 = 7. This verifies the procedure of evaluating a function by direct substitution.
Solve the absolute value equation: |x - 3| = 4.
x = -7 or x = 3
x = -1 or x = 7
x = 1 or x = 7
x = -3 or x = 4
The absolute value equation splits into two cases: x - 3 = 4 and x - 3 = -4, resulting in x = 7 and x = -1 respectively. Both solutions are valid unless excluded by context.
Solve the system of equations: 2x + y = 7 and x - y = 1.
x = 2, y = 3
x = 4, y = 3
x = 3, y = 1
x = 8/3, y = 5/3
By expressing y from the second equation as y = x - 1 and substituting into the first, the equation simplifies to 3x - 1 = 7. Solving this yields x = 8/3 and consequently y = 5/3.
Simplify the expression: (x² - 9)/(x - 3).
x - 3
x² - 3
1
x + 3
The numerator factors as (x - 3)(x + 3), which cancels with the denominator (x - 3), leaving x + 3. This cancellation is valid as long as x ≠3.
Solve for x: 2^(x) = 16.
16
8
4
2
Since 16 is 2 raised to the power of 4, the equation 2^(x) = 2^(4) implies x = 4. Equating exponents with the same base confirms the answer.
Solve the quadratic equation: x² - 5x + 6 = 0.
x = 2 or x = 3
x = 3
x = -2 or x = -3
x = 1
Factoring the quadratic gives (x - 2)(x - 3) = 0, so the solutions are x = 2 and x = 3. Each factor equal to zero confirms the distinct roots.
If f(x) = x² - 4x + 3, factor the function and determine its zeros.
x = 2 or x = 4
x = -1 or x = -3
x = 1 or x = 3
x = 1
The quadratic factors as (x - 1)(x - 3), indicating the zeros occur when x = 1 or x = 3. This factorization method is a fundamental tool for finding the roots of polynomials.
Solve the radical equation: √(3x + 1) = 4.
5
16
4
15
Squaring both sides yields 3x + 1 = 16, which simplifies to 3x = 15 and therefore x = 5. It is essential to verify the solution in the original equation to rule out extraneous roots.
Simplify the rational expression: (2x² - 8)/(2x).
2(x - 2)(x + 2)/x
(x - 2)/2x
(x² - 4)/x
(x - 2)(x + 2)/x
Begin by factoring 2x² - 8 as 2(x² - 4) and recognizing that x² - 4 factors into (x - 2)(x + 2). Canceling the common factor of 2 in the numerator with the denominator leads to (x - 2)(x + 2)/x.
Solve for x in the equation: (x - 2)/(x + 3) = 2/5.
16/5
16/3
14/3
10/3
Cross-multiplying gives 5(x - 2) = 2(x + 3), which expands to 5x - 10 = 2x + 6. Solving for x results in 3x = 16, so x = 16/3. This method emphasizes the reliability of cross-multiplication in rational equations.
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Study Outcomes

  1. Apply algebraic properties to simplify expressions and solve linear equations.
  2. Analyze algebraic problems and determine appropriate strategies for solution.
  3. Evaluate systems of equations using substitution and elimination methods.
  4. Demonstrate proficiency in factoring polynomials and simplifying algebraic expressions.
  5. Assess problem-solving skills to confidently approach exam-style questions.

Intermediate Algebra Practice Test Cheat Sheet

  1. Linear Equations & Inequalities - Dive into solving one‑variable equations and inequalities, even when letters sneak in as mysterious coefficients. Picture each equation as a balance scale waiting for you to tip it just right! Mastering this unlocks the door to more advanced algebra puzzles. thecorestandards.org
  2. Remainder Theorem - Discover how plugging a number into a polynomial instantly gives you the remainder when dividing by (x − a). This shortcut transforms long division into a quick mental calculation! It's your secret weapon for factoring and checking roots fast. thecorestandards.org
  3. Quadratic Equations - Explore multiple ways to crack quadratics: factoring, completing the square, and the trusty quadratic formula. Learn to spot when solutions go beyond the real world into a ± bi territory. By the end, you'll handle every parabola like a pro! thecorestandards.org
  4. Polynomial Arithmetic - Get hands‑on with adding, subtracting, and multiplying polynomials - think of them as algebraic recipes where like terms blend perfectly. Understanding closure under these operations means you won't get lost mixing terms. It's math magic at its finest. thecorestandards.org
  5. Systems of Linear Equations - Crack the code of two or more equations working together by graphing, substitution, or elimination methods. It's like solving a detective mystery where each variable leaves clues on the grid. Soon you'll juggle multiple equations without breaking a sweat! thecorestandards.org
  6. Expression Structure - Break expressions into terms, factors, and coefficients as if you're dissecting a math puzzle. Understanding this anatomy helps you simplify and rewrite expressions in a flash. Think of it as getting to know the LEGO pieces before building your algebra tower. thecorestandards.org
  7. Zeros & Factors of Polynomials - Link the roots of a polynomial to its factors and use this to sketch rough graphs. Hitting each zero tells you where the curve crosses the x‑axis - vital for visualizing behavior. With practice, you'll predict shapes like a graphing wizard! thecorestandards.org
  8. Modeling with Equations & Inequalities - Turn real‑world scenarios into one‑variable equations or inequalities, whether linear or quadratic. From budget planning to physics problems, you'll learn to wield algebra like a language of the universe. It's math that makes sense outside the classroom! thecorestandards.org
  9. Binomial Theorem - Master the art of expanding (x + y)n using Pascal's Triangle or combinatorial coefficients. This theorem turns massive expansions into a patterned breeze. Perfect for tackling advanced problems in algebra and beyond! thecorestandards.org
  10. Rational Expression Rewrite - Practice expressing a(x)/b(x) as q(x) + r(x)/b(x) by performing polynomial long division. This skill demystifies complex fractions and makes solving rational equations far more approachable. Soon, you'll simplify like a champ! thecorestandards.org
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