GRE Algebra Practice Questions Practice Questions with Answers

Algebra accounts for approximately 30% of the Quantitative Reasoning section on the GRE, requiring mastery of variables, expressions, and equations.

Success on this exam involves more than just memorizing formulas; it requires the ability to manipulate symbols and solve complex word problems under time pressure. Whether you are dealing with linear equations or quadratic functions, a systematic approach is essential for accuracy. To build a strong foundation, many students start with GRE Arithmetic Practice Questions before moving into more abstract algebraic concepts. This guide provides comprehensive GRE Algebra Practice Questions and detailed explanations to help you refine your skills and boost your confidence for test day.

Concept Explanation

GRE algebra is the study of mathematical symbols and the rules for manipulating these symbols to solve for unknown variables. At its core, this topic covers four major areas: simplifying algebraic expressions, solving linear and quadratic equations, working with inequalities, and interpreting coordinate geometry. On the GRE, you will encounter these concepts in multiple formats, including Quantitative Comparison, Multiple Choice, and Numeric Entry.

To excel, you must be comfortable with the following sub-topics:

• Linear Equations: Solving for a single variable $x$ or systems of equations with two variables.
• Quadratic Equations: Factoring polynomials or using the quadratic formula to find roots of equations such as $$ax^2 + bx + c = 0$$
• Inequalities: Understanding how to flip the inequality sign when multiplying or dividing by a negative number.
• Functions: Evaluating $f(x)$ and understanding domain and range.

If you are looking for a structured way to manage your study time across these topics, using an AI MasterPlan can help you organize your daily practice sessions effectively. For those who have already mastered the basics, moving on to GRE Data Analysis Questions will round out your quantitative preparation.

Solved Examples

Below are three fully worked examples demonstrating common algebraic manipulations found on the GRE.

1. Solving for a Variable: Solve for $x$ in the equation $$3(x - 4) = 2x + 7$$
   1. Distribute the 3 on the left side: $3x - 12 = 2x + 7$.
   2. Subtract $2x$ from both sides to isolate the variable: $x - 12 = 7$.
   3. Add 12 to both sides: $x = 19$.
   4. The value of $x$ is 19.
2. Quadratic Factoring: Find the roots of the equation $$x^2 - 5x + 6 = 0$$
   1. Identify two numbers that multiply to 6 and add to -5. These are -2 and -3.
   2. Write the equation in factored form: $(x - 2)(x - 3) = 0$.
   3. Set each factor to zero: $x - 2 = 0$ or $x - 3 = 0$.
   4. The roots are $x = 2$ and $x = 3$.
3. System of Equations: Solve for $y$ given $$2x + y = 10$$ and $$x - y = 2$$
   1. Use the elimination method by adding the two equations together: $(2x + y) + (x - y) = 10 + 2$.
   2. The $y$ terms cancel out: $3x = 12$.
   3. Divide by 3: $x = 4$.
   4. Substitute $x = 4$ back into the second equation: $4 - y = 2$.
   5. Solve for $y$: $y = 2$.

Practice Questions

1. If $5x - 3 = 2x + 9$, what is the value of $x$?

2. Solve the inequality for $z$: $$-4z + 8 < 20$$

3. Factor the expression $x^2 - 16$ completely.

4. If $f(x) = 2x^2 - 3x + 1$, what is the value of $f(-2)$?

5. Quantity A: $(x + y)^2$
Quantity B: $x^2 + y^2$
Given that $x > 0$ and $y > 0$, which quantity is greater?

6. Solve for $a$ in the equation $$\frac{a}{4} + \frac{a}{3} = 7$$

7. A rectangle has a perimeter of 40 units. If the length is $3x$ and the width is $x + 2$, find the value of $x$.

8. Simplify the expression: $$\frac{12x^3y^2}{4xy}$$

9. If $2^n = 64$, what is the value of $n + 2$?

10. Solve the system for $x$:
$$3x + 2y = 16$$
$$x - 2y = 0$$

Answers & Explanations

1. Answer: 4. Subtract $2x$ from both sides to get $3x - 3 = 9$. Add 3 to both sides to get $3x = 12$. Dividing by 3 yields $x = 4$.
2. Answer: $z > -3$. Subtract 8 from both sides: $-4z < 12$. When dividing by -4, you must flip the inequality sign, resulting in $z > -3$.
3. Answer: $(x - 4)(x + 4)$. This is a difference of squares, which follows the pattern $a^2 - b^2 = (a - b)(a + b)$. Here, $a = x$ and $b = 4$.
4. Answer: 15. Plug -2 into the function: $2(-2)^2 - 3(-2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15$.
5. Answer: Quantity A. Expanding Quantity A gives $x^2 + 2xy + y^2$. Since $x$ and $y$ are positive, $2xy$ must be positive, making Quantity A larger than $x^2 + y^2$.
6. Answer: 12. Find a common denominator (12): $\frac{3a}{12} + \frac{4a}{12} = 7$. This simplifies to $\frac{7a}{12} = 7$. Multiplying by 12 and dividing by 7 gives $a = 12$.
7. Answer: 4.5. Perimeter $P = 2(L + W)$. So, $40 = 2(3x + x + 2)$. Simplify: $40 = 2(4x + 2) \rightarrow 40 = 8x + 4$. Subtract 4: $36 = 8x$. Divide by 8: $x = 4.5$.
8. Answer: $3x^2y$. Divide the coefficients: $12/4 = 3$. Subtract exponents for like bases: $x^{3-1} = x^2$ and $y^{2-1} = y$.
9. Answer: 8. Since $2^6 = 64$, $n = 6$. Therefore, $n + 2 = 8$.
10. Answer: 4. Add the two equations: $(3x + 2y) + (x - 2y) = 16 + 0$. This gives $4x = 16$, so $x = 4$.

For more practice on related topics, you might explore GRE Statistics Practice Questions or use the AI Question Generator to create custom algebra drills.

1. Which of the following is equivalent to \( (x - 3)^2 \)?

• A\( x^2 - 9 \)
• B\( x^2 + 9 \)
• C\( x^2 - 6x + 9 \)
• D\( x^2 - 3x + 9 \)

Frequently Asked Questions

How much algebra is on the GRE?

Algebra typically makes up about one-third of the Quantitative Reasoning section, covering topics from simple linear equations to coordinate geometry. It is one of the most heavily tested areas alongside arithmetic and data analysis.

Do I need to memorize the quadratic formula for the GRE?

While the GRE rarely requires the full quadratic formula, knowing it is helpful for equations that cannot be easily factored. Most GRE quadratics are designed to be solved via factoring or by recognizing common algebraic patterns like the difference of squares.

How do I handle inequalities with negative numbers?

The most important rule for inequalities is to reverse the direction of the inequality sign whenever you multiply or divide both sides by a negative number. Failing to do this is a common source of errors on the exam.

What is the difference between an expression and an equation?

An expression is a mathematical phrase containing numbers and variables, such as $3x + 5$, while an equation states that two expressions are equal, such as $3x + 5 = 11$. You simplify expressions, but you solve equations.

Are calculators allowed for algebra questions on the GRE?

Yes, an on-screen calculator is provided for the Quantitative Reasoning section. However, it is often faster to solve algebraic problems manually through simplification rather than relying on the calculator for every step.