Hard GRE Algebra Practice Questions Practice Questions

Algebraic manipulation accounts for nearly 40% of the Quantitative Reasoning section, making it the most significant mathematical domain for test-takers. These hard GRE algebra practice questions focus on the upper echelon of difficulty, where simple arithmetic gives way to abstract reasoning, multi-step inequalities, and complex functions. To succeed on the GRE Prep journey, you must move beyond basic variable solving and begin recognizing patterns in quadratic forms and coordinate geometry.

Concept Explanation

Hard GRE algebra practice questions involve high-level manipulation of variables, exponents, and functions to solve for unknown values or compare quantities. At this level, the test shifts from testing basic rules to testing your ability to apply those rules in non-obvious ways. Key concepts include quadratic equations that require factoring or the quadratic formula, absolute value inequalities with multiple solutions, and functions defined by custom operations. You will also encounter GRE practice questions with explanations that utilize the properties of exponents and roots to simplify seemingly impossible expressions. Success depends on maintaining algebraic balance—whatever operation you perform on one side of an equation must be performed on the other—while strategically substituting values to test the limits of inequalities.

Solved Examples

Reviewing these step-by-step solutions will help you internalize the logic required for the most difficult algebra problems on the exam.

1. Example 1: Quadratic Inequalities
   Solve for $x$: $$x^2 - 5x + 6 < 0$$
   1. Factor the quadratic expression: $(x - 2)(x - 3) < 0$.
   2. Identify the critical points where the expression equals zero: $x = 2$ and $x = 3$.
   3. Test the intervals created by these points: $(-\infty, 2)$, $(2, 3)$, and $(3, \infty)$.
   4. For $x = 2.5$ (in the middle interval), $(2.5 - 2)(2.5 - 3) = (0.5)(-0.5) = -0.25$, which is less than zero.
   5. The solution is $2 < x < 3$.
2. Example 2: Simultaneous Equations with Three Variables
   If $a + b = 7$, $b + c = 10$, and $a + c = 11$, what is the value of $a + b + c$?
   1. Add all three equations together: $(a + b) + (b + c) + (a + c) = 7 + 10 + 11$.
   2. Simplify the left side: $2a + 2b + 2c = 28$.
   3. Factor out the 2: $2(a + b + c) = 28$.
   4. Divide by 2 to find the sum: $a + b + c = 14$.
3. Example 3: Function Notation
   Let the function $f$ be defined by $f(x) = x^2 - \frac{1}{x}$. What is the value of $f(f(1))$?
   1. First, find $f(1)$: $f(1) = 1^2 - \frac{1}{1} = 1 - 1 = 0$.
   2. Now, find $f(0)$ using the original function definition.
   3. Notice that $f(0) = 0^2 - \frac{1}{0}$ involves division by zero.
   4. In GRE contexts, functions are only defined where the denominator is non-zero; therefore, this specific value is undefined.

Practice Questions

1. If $3^{x+1} = 9^{x-2}$, what is the value of $x$?
2. Find the value of $y$ if $\frac{2}{y} + \frac{3}{2y} = 14$.
3. Quantity A: $|x + 3|$
   Quantity B: $|x| + 3$
   Compare the two quantities if $x$ is any real number.

4. If $x^2 - y^2 = 24$ and $x - y = 4$, what is the value of $xy$?
5. A line in the $xy$-plane passes through the points $(2, 5)$ and $(4, k)$. If the slope of the line is $-3$, what is the value of $k$?
6. If $f(x) = 2x - 3$ and $g(x) = x^2 + 1$, what is the value of $g(f(2))$?
7. Solve for $z$ in the equation: $$\sqrt{z + 5} - \sqrt{z} = 1$$
8. If $a$ and $b$ are positive integers such that $a^b = 64$, how many possible values are there for $a$?
9. Quantity A: $(x + y)^2$
   Quantity B: $x^2 + y^2$
   Compare the two quantities if $xy > 0$.
10. If $\frac{x}{y} = \frac{2}{3}$ and $\frac{y}{z} = \frac{6}{5}$, what is the value of $\frac{z}{x}$?

Answers & Explanations

1. Answer: 5. Rewrite 9 as $3^2$. The equation becomes $3^{x+1} = (3^2)^{x-2}$, which simplifies to $3^{x+1} = 3^{2x-4}$. Since the bases are equal, the exponents must be equal: $x + 1 = 2x - 4$. Solving for $x$ gives $x = 5$.
2. Answer: 1/4. Find a common denominator for the fractions: $\frac{4}{2y} + \frac{3}{2y} = \frac{7}{2y}$. Set this equal to 14: $\frac{7}{2y} = 14$. Multiply both sides by $2y$: $7 = 28y$. Divide by 28: $y = \frac{7}{28} = \frac{1}{4}$.
3. Answer: Quantity B is greater than or equal to Quantity A. By the Triangle Inequality theorem in algebra, $|a + b| \leq |a| + |b|$. If $x$ is positive, the quantities are equal. If $x$ is negative (e.g., $-5$), Quantity A is $|-2| = 2$ and Quantity B is $|-5| + 3 = 8$. Thus, B is never smaller than A.
4. Answer: 5. Use the difference of squares: $x^2 - y^2 = (x - y)(x + y)$. Substitute the known values: $24 = 4(x + y)$, so $x + y = 6$. We now have a system: $x - y = 4$ and $x + y = 6$. Adding them gives $2x = 10$, so $x = 5$. Subtracting them gives $2y = 2$, so $y = 1$. Finally, $xy = 5 \times 1 = 5$.
5. Answer: -1. Use the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. Here, $-3 = \frac{k - 5}{4 - 2}$. This simplifies to $-3 = \frac{k - 5}{2}$. Multiply by 2: $-6 = k - 5$. Add 5 to both sides: $k = -1$.
6. Answer: 2. First find $f(2) = 2(2) - 3 = 1$. Then substitute this into $g(x)$: $g(1) = 1^2 + 1 = 2$.
7. Answer: 4. Isolate one radical: $\sqrt{z + 5} = 1 + \sqrt{z}$. Square both sides: $z + 5 = (1 + \sqrt{z})^2 = 1 + 2\sqrt{z} + z$. Subtract $z$ and 1 from both sides: $4 = 2\sqrt{z}$. Divide by 2: $2 = \sqrt{z}$. Square again: $z = 4$.
8. Answer: 3. List the pairs of $(a, b)$ that result in 64: $(64, 1)$, $(8, 2)$, $(4, 3)$, $(2, 6)$. The possible values for $a$ are 64, 8, 4, and 2. However, the question asks for the number of values. There are 4 possible values.
9. Answer: Quantity A is greater. Expand Quantity A: $(x + y)^2 = x^2 + 2xy + y^2$. Quantity B is $x^2 + y^2$. Since the problem states $xy > 0$, the term $2xy$ must be positive. Therefore, A will always be larger than B by the amount $2xy$.
10. Answer: 5/4. Multiply the two ratios: $\frac{x}{y} \times \frac{y}{z} = \frac{2}{3} \times \frac{6}{5}$. This simplifies to $\frac{x}{z} = \frac{12}{15} = \frac{4}{5}$. The question asks for $\frac{z}{x}$, which is the reciprocal: $\frac{5}{4}$.

1. If \( 2x - 5 = 11 \), what is the value of \( x^2 \)?

• A64
• B16
• C36
• D49

Frequently Asked Questions

How many algebra questions are on the GRE?

Algebra typically makes up about 30% to 40% of the Quantitative Reasoning section, covering topics from linear equations to coordinate geometry. It is one of the four major content areas alongside arithmetic, geometry, and data analysis.

What is the best way to handle complex GRE inequalities?

The most effective strategy is to treat the inequality like an equation to find critical points, then test values in each resulting interval. Remember to flip the inequality sign whenever you multiply or divide both sides by a negative number.

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

While most GRE quadratics can be solved via factoring, knowing the quadratic formula is a helpful backup for difficult polynomials. Most test-takers find that practicing unlimited GRE practice questions helps them recognize which method is faster for a given problem.

How does coordinate geometry relate to algebra on the test?

Coordinate geometry is essentially the visual representation of algebra, where lines and curves are defined by equations like $y = mx + b$. Harder questions often require you to find the intersection of two lines by solving a system of linear equations.

Can I use a calculator for algebra problems on the GRE?

Yes, an on-screen calculator is provided, but it is often faster to solve algebraic expressions through simplification and factoring. Relying too heavily on the calculator for variables can lead to time management issues during the exam.