GRE Algebra Practice Test Practice Questions with Answers

Concept Explanation

GRE algebra involves the manipulation of symbols and equations to solve for unknown variables in linear, quadratic, and inequality formats. Success on the Quantitative Reasoning section requires a firm grasp of algebraic expressions, functions, and coordinate geometry. At its core, algebra is about maintaining balance; whatever operation you perform on one side of an equation, you must perform on the other. This consistency allows you to isolate a variable like $x$ or $y$ and determine its value. For many students, the GRE Prep process involves revisiting these fundamental rules and applying them to complex word problems and Quantitative Comparison formats. According to ETS, the makers of the GRE, the test assesses your ability to model real-world scenarios using algebraic relationships. This includes understanding exponents, roots, and the properties of functions.

Solved Examples

1. Linear Equation: Solve for $x$ in the equation $3(x - 4) = 2x + 7$.
   1. Distribute the 3 into the parentheses: $3x - 12 = 2x + 7$.
   2. Subtract $2x$ from both sides to group variables: $x - 12 = 7$.
   3. Add 12 to both sides to isolate $x$: $x = 19$.
2. System of Equations: If $2x + y = 10$ and $x - y = 2$, find the value of $x$.
   1. Use the elimination method by adding the two equations together: $(2x + y) + (x - y) = 10 + 2$.
   2. Simplify the left side (the $y$ terms cancel out): $3x = 12$.
   3. Divide by 3: $x = 4$.
3. Quadratic Factoring: Find the roots of the equation $x^2 - 5x + 6 = 0$.
   1. Look for two numbers that multiply to 6 and add to -5. Those numbers are -2 and -3.
   2. Rewrite 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 solutions are $x = 2$ and $x = 3$.
4. Inequalities: Solve for $z$ in the inequality $-2z + 5 < 13$.
   1. Subtract 5 from both sides: $-2z < 8$.
   2. Divide by -2. Remember that dividing by a negative number flips the inequality sign: $z > -4$.

Practice Questions

1. If $\frac{x}{3} + 4 = 10$, what is the value of $x^2$?

2. Solve for $a$ in the equation: $5a - 2(a + 3) = 12$.

3. Quantitative Comparison:
Quantity A: $(x + 2)^2$
Quantity B: $x^2 + 4x + 4$

4. If $3x - 2y = 8$ and $y = 2x - 5$, what is the value of $x$?

5. Simplify the expression: $\frac{16x^4y^2}{4x^2y}$, assuming $x, y \neq 0$.

6. Solve the inequality: $4 - 3x \geq 19$.

7. If the function $f(x) = x^2 - 3x + 1$, what is $f(-2)$?

8. A rental car costs $40 per day plus $0.15 per mile driven. If a customer paid $73 for a one-day rental, how many miles did they drive? Use AI MasterPlan to schedule your practice for word problems like this.

9. Factor the expression completely: $3x^2 - 12$.

10. If $x$ is an integer and $2 < 2x - 4 < 10$, what are the possible values of $x$?

Answers & Explanations

1. Answer: 324. First, solve for $x$: Subtract 4 from both sides to get $\frac{x}{3} = 6$. Multiply by 3 to find $x = 18$. The question asks for $x^2$, so $18^2 = 324$.
2. Answer: 6. Distribute the -2: $5a - 2a - 6 = 12$. Combine like terms: $3a - 6 = 12$. Add 6: $3a = 18$. Divide by 3: $a = 6$.
3. Answer: The two quantities are equal. Expanding Quantity A using the FOIL method: $(x + 2)(x + 2) = x^2 + 2x + 2x + 4 = x^2 + 4x + 4$. This matches Quantity B exactly.
4. Answer: 2. Substitute the second equation into the first: $3x - 2(2x - 5) = 8$. Distribute: $3x - 4x + 10 = 8$. Combine terms: $-x + 10 = 8$. Subtract 10: $-x = -2$, so $x = 2$.
5. Answer: $4x^2y$. Divide the coefficients: $16/4 = 4$. Subtract the exponents for $x$: $4 - 2 = 2$. Subtract the exponents for $y$: $2 - 1 = 1$. Result: $4x^2y$.
6. Answer: $x \leq -5$. Subtract 4 from both sides: $-3x \geq 15$. Divide by -3 and reverse the inequality sign: $x \leq -5$.
7. Answer: 11. Substitute -2 for $x$: $(-2)^2 - 3(-2) + 1 = 4 + 6 + 1 = 11$.
8. Answer: 220 miles. Let $m$ be miles. The equation is $40 + 0.15m = 73$. Subtract 40: $0.15m = 33$. Divide by 0.15: $m = 220$.
9. Answer: $3(x - 2)(x + 2)$. First, factor out the greatest common factor, 3: $3(x^2 - 4)$. Recognize that $x^2 - 4$ is a difference of squares: $3(x - 2)(x + 2)$.
10. Answer: 4, 5, and 6. Add 4 to all parts of the inequality: $6 < 2x < 14$. Divide all parts by 2: $3 < x < 7$. Since $x$ is an integer, the values are 4, 5, and 6.

1. Which of the following is equivalent to the expression \( (2x - 3)(x + 4) \)?

• A\( 2x^2 + 5x - 12 \)
• B\( 2x^2 - 12 \)
• C\( 2x^2 + 11x - 12 \)
• D\( 2x^2 + 5x + 12 \)

Frequently Asked Questions

What algebra topics are most common on the GRE?

The GRE focuses heavily on linear equations, systems of equations, inequalities, and basic quadratic factoring. You will also frequently encounter coordinate geometry and the interpretation of algebraic functions in word problems.

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

While the quadratic formula is useful, most quadratic equations on the GRE can be solved more quickly through factoring or by plugging in answer choices. However, knowing it can provide a helpful safety net for complex problems.

How should I handle Quantitative Comparison algebra questions?

For these questions, try to simplify both expressions as much as possible before comparing. If the relationship depends on whether a variable is positive, negative, or a fraction, test numbers from different categories to see if the relationship changes.

What is the best way to avoid mistakes with negative signs?

Always use parentheses when substituting negative numbers into an expression and remember to flip the inequality sign when multiplying or dividing by a negative value. Slowing down during distribution steps can prevent the most common calculation errors.

Are calculators allowed for algebraic computations on the GRE?

Yes, an on-screen calculator is provided for the Quantitative Reasoning section. However, it is often faster to simplify algebraic expressions manually than to rely on the calculator for every step.