Comprehensive GRE Algebra Study Guide

The Importance of Algebra on the Revised GRE

• Algebra is considered one of the most critical subjects for the Revised GRE, appearing directly in approximately $10\,\%$ to $15\,\%$ of all mathematics questions.
• Beyond direct testing, Algebra is the foundational skill required to solve Word Problems ($33\,\%$ of the exam), Geometry ($20\,\%$), and Charts and Graphs ($15\,\%$).
• Algebra is described as the "heart" of GRE math; without it, more exotic concepts cannot be solved.
• Prerequisite topics for mastering Algebra include PEMDAS, Arithmetic, Fractions, Decimals, Integer Properties, Exponents, and Roots.

Solving for Variables

• The primary objective of most algebraic equations is to solve for a variable ($x$, $y$, etc.), which represents an unknown value.
• A standard three-step process is recommended for isolating a variable:
  • Identify what needs to be removed to isolate the variable.
  • Remove that element by performing the opposite operation (e.g., the opposite of addition is subtraction; the opposite of multiplication is division).
  • Perform the identical operation to both sides of the equal sign to maintain balance.
• Equations vs. Expressions:
  • An equation contains an equal sign ($=$, e.g., $3x - 6 = 9$). Equations can be solved to find specific values.
  • An expression does not have an equal sign (e.g., $4x + 7$). Its value is dependent on the number plugged into the variable.
• To isolate a variable, apply PEMDAS in reverse:
  • First, use addition and subtraction to move terms.
  • Second, use multiplication and division to move coefficients.
  • Third, address exponents or roots and resolve parentheses.
• Validation: Always plug the resulting solution back into the original equation to ensure it results in a true statement.

Grouping Like Terms, Distribution, and Factoring

• Like terms share the same variable and the same exponent (e.g., $5x$ and $2x$ or $4x^{2}$ and $x^{2}$). Terms with different variables or differing exponents are not like terms.
• Distribution involves multiplying a term outside parentheses with every element inside the parentheses. Example: $4(3n + 5) = 12n + 20$.
• Watch for negative sign distribution: $4 - (3x - 1)$ simplifies to $4 - 3x + 1$, or $5 - 3x$.
• Factoring is distribution in reverse, splitting an expression into simpler terms.
• When factoring intimidating expressions, resolve two questions:
  • What is the largest element common to all terms?
  • What remains when that common element is divided out?
• Example: In the expression $12n^{3} + 4n^{2} + 8n$, the largest common element is $4n$. Factored, this becomes $4n(3n^{2} + n + 2)$.

The Denominator Trick and Reciprocals

• Equations with denominators are common sources of errors. To eliminate them, use the "Denominator Trick": immediately multiply the entire equation by the denominator you wish to remove.
• If multiple denominators are present, multiply the entire equation by the Lowest Common Denominator (LCD).
• Cross-Multiplication: If a single fraction equals another single fraction (e.g., $\frac{a}{b} = \frac{c}{d}$), multiply each denominator with the opposite numerator ($ad = bc$).
• If either side of the equation is not a fraction, do not cross-multiply; use the Denominator Trick instead.
• Reciprocal Shortcut: If a variable has a fractional coefficient (e.g., $\frac{2}{3}n = 4$), multiply both sides by the reciprocal ($n = 4 \times \frac{3}{2} = 6$).

Variables "In Terms Of" Each Other

• Problems asking for "Variable A in terms of Variable B" require solving for Variable A while leaving Variable B in the final answer.
• The "Question Variable" rule: Always solve for the variable that follows the words "what is" or appears before "in terms of."
• Never solve for the variable mentioned after "in terms of"; that variable should appear in the answer choices.
• Use the Complex Numerator Shortcut to simplify results: $\frac{a + b}{c} = \frac{a}{c} + \frac{b}{c}$.

Inequalities

• Inequality symbols include:
  • $x < 3$ ($x$ is less than $3$).
  • $x > 5$ ($x$ is greater than $5$).
  • $x \leq 4$ ($x$ is less than or equal to $4$).
  • $x \geq 8$ ($x$ is greater than or equal to $8$).
• Inequalities are solved like equations with one critical exception: Flip the direction of the inequality sign when multiplying or dividing by a negative number.
• Dividing or multiplying by a variable is dangerous because its sign is unknown. If the sign is unknown, you must account for two scenarios: one where the variable is positive and one where it is negative.
• Compound Inequalities ($a < x < b$): Isolate the variable in the middle by performing the same operation on all three sections of the inequality.

Absolute Value

• Absolute value ($|x|$) represents the distance on a number line between a value and zero.
• Solving Absolute Value Equations:
  • First, isolate the absolute value brackets.
  • Drop the brackets and place a plus/minus ($\pm$) sign in front of the algebraic expression: $\pm(\text{Algebra}) = \text{Answer}$.
  • Solve both resulting equations.
  • Plug solutions back into the original equation to check for validity; some solutions may be extraneous.
• Absolute Value Inequalities: Follow the same isolation and $\pm$ steps. Solve both, then combine the results into a single range statement.

Systems of Equations

• Substitution: Isolate one variable in one equation and plug that expression into the second equation.
• Elimination: Stack the equations, align the variables, and add or subtract them to cancel one variable. This is usually the faster method.
• Opposite Coefficients: For elimination to work, the targeted variable must have opposite coefficients (e.g., $-3y$ and $+3y$).
• "Solving for Chunks": Sometimes, the GRE does not require solving for individual variables. You might be able to add or subtract equations to find the value of a larger expression (the "chunk") directly.

Exponents and Roots in Algebra

• To solve for a variable under a root, isolate the root and then raise both sides to the power matching the root's degree.
• To solve for a variable raised to a power, isolate the exponent and take the corresponding root of both sides.
• Even Exponents: Generally yield two solutions (one positive, one negative). Example: $x^{2} = 25$ means $x = 5$ or $x = -5$.
• Odd Exponents: Yield exactly one solution. Example: $x^{3} = -27$ means $x = -3$.
• Technical Rule: Taking the root of an even exponent technically results in the absolute value ($\sqrt{n^{2}} = |n|$).

Quadratic Equations and FOIL

• A quadratic equation contains a variable raised to both the first and second powers ($ax^{2} + bx + c = 0$).
• Factoring quadratics: Set the equation to zero, ensure the $x^{2}$ term is positive, and find two numbers that multiply to the constant ($c$) and add to the middle coefficient ($b$).
• Setting factors to zero: If $(x + 2)(x + 5) = 0$, then $x = -2$ or $x = -5$. These solutions are also called the "roots" of the equation.
• FOIL (First, Outer, Inner, Last): A mnemonic for multiplying two factors.
• The three mandatory FOIL Identities:
  • $(x + y)^{2} = x^{2} + 2xy + y^{2}$
  • $(x - y)^{2} = x^{2} - 2xy + y^{2}$
  • $x^{2} - y^{2} = (x + y)(x - y)$.

Quantitative Comparison: Plug-Ins and Manipulation

• "Plug-In" problems involve comparing expressions rather than solving equations.
• Strategy: Plug in numbers to try to prove an inconsistent relationship (Answer D).
• Testing Criteria: Always test at least three different types of numbers ($1$, $0$, and $-1$) because they reveal unexpected results.
• Choosing combinations: If two variables are used, try different combinations of $1$, $0$, and $-1$.
• Extreme Values: Consider fractions between $0$ and $1$ (which grow larger when rooted and smaller when squared), or very large/small numbers.
• Term Removal: Remove any terms shared by both Quantity A and Quantity B before plugging to simplify the comparison.
• Manipulation Rules: You can add/subtract from both sides, multiply/divide by a positive value, or square/root both sides if both quantities are positive.

Advanced Algebraic Concepts

• Shortcut ($x - 1$ vs. $1 - x$): Recognizing that $1 - x = -1(x - 1)$ can simplify complex fractions.
• "Something" Shortcut: Simplify nested or complex equations by treating a part as a single unit or "something." Equation: $6 - \text{something} = 4$. Therefore, the "something" must equal $2$.
• Avoiding Quadratics: If you have a sum ($x + y$) and a product ($xy$), find the two numbers that fit both logically to avoid factoring a full quadratic.
• Maximization Problems: To find the greatest or smallest value of an expression, identify the endpoints of the input variables' ranges and test every combination of those endpoints.
• Integer Constraints (Addition/Subtraction Multiple Rule): If two terms in an equation are multiples of $M$, the third term must also be a multiple of $M$ for integer solutions to exist.
• Absolute Value Graph Formula: To translate a number line into an inequality, use:
  • $|x - \text{Midpoint}| \leq \text{Distance from Endpoints to Midpoint}$.