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GRE Algebra Notes, Formulas, Strategy, and Practice

Build a GRE-ready algebra toolkit with original notes on equations, inequalities, exponents, roots, factoring, quadratics, functions, word problems, and Quantitative Comparison.

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What a strong GRE tutor wants you to master here

This is the algebra layer that keeps showing up in real GRE Quant work: not just solving equations, but choosing the shortest valid move, reading restrictions early, and knowing when testing values beats grinding through symbols.

Algebra You Can Reuse

Linear equations, expression targeting, simplification, and the restrictions that quietly control the answer.

Inequalities That Behave

Sign reversals, interval testing, compound ranges, and absolute-value cases without guesswork.

Power, Roots, and Factoring

Exponent rules, radical cleanup, quadratic structure, and the factoring patterns GRE keeps recycling.

Tutor-Level Decision Making

When to solve, when to test values, when to target an expression, and how to avoid QC traps.

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GRE Intro

How to Think About GRE Algebra

Treat GRE algebra as decision-making, not just manipulation. Start with the structure, write the restrictions, choose the shortest legal move, and only then calculate.

1. How to Use This GRE Algebra Guide

Work this page in layers. On the first pass, focus on the decision rules and the logic behind each move. On the second pass, cover the explanations and solve the examples yourself. On the third pass, use the diagnostic and review blocks for timed practice.

GRE habit: ask what the question wants before solving anything. Many GRE algebra questions ask for an expression, comparison, or range rather than a single variable.

If a question looks long, reduce it to structure: equation, inequality, expression target, factor pattern, or test-values problem.

2. GRE Algebra Map

Most GRE algebra questions fall into a small set of reusable families. Recognizing the family often matters more than calculating quickly.

Family	Core Move	Typical Trap
Linear equations	Isolate the requested quantity	Solving more than needed
Inequalities	Track sign direction carefully	Forgetting reversal after division by a negative
Factored expressions	Use zero-product or cancel legal factors	Cancelling terms instead of factors
Quadratics	Move to zero, then factor or compare roots	Losing a root by dividing too early
Quantitative Comparison	Test more than one legal case	Proving from only one example

3. Linear Equations and Expression Targeting

Linear equations are usually the fastest GRE algebra questions, but the exam often hides a shortcut. If the prompt asks for a new expression built from the one you already know, scale the equation instead of solving for individual variables.

Targeting rule: if you know

ax + by

and the question asks for

k(ax + by) + c

, transform directly.

Original example: If

4m - 3n = 19

, then

8m - 6n + 5 = 2(19) + 5 = 43

That approach is both faster and safer than trying to solve for $m$ and $n$ from one equation.

4. Restrictions, Denominators, and Legal Values

Before simplifying any algebraic fraction or radical, write the legal restrictions. GRE questions often reward students who notice where an expression stops being defined.

A denominator cannot be zero.
The expression inside an even root must be nonnegative.
The principal square root is never negative.

Trap: after simplifying

(x^2-9)/(x+3)

x-3

, the original restriction

x \ne -3

still remains.

5. Inequalities and Boundary Thinking

Treat inequalities like equations with one extra rule: multiplying or dividing by a negative reverses the sign.

-5x > 20 \Rightarrow x < -4

For product inequalities such as $(x-2)(x+5) > 0$ , first mark the boundary points where the expression becomes zero. Then test one value from each interval.

Interval rule: boundary points split the number line into zones. Test zones, not random values.

6. Absolute Value as Distance

Absolute value measures distance from zero, so it is never negative. That turns many GRE questions into clean two-case or interval problems.

Form	Meaning	Result
$\|x\| = a$	Distance from zero is exactly $a$	$x = \pm a$
$\|x\| < a$	x lies within $a$ units of zero	$-a < x < a$
$\|x\| > a$	x lies more than $a$ units from zero	$x < -a$ or $x > a$

7. Exponents, Roots, and Rewrite Strategy

Exponent and radical questions become easier when you rewrite everything in the same base or same factor structure.

$a^m \cdot a^n = a^{m+n}$
$a^{-n} = 1/a^n$
$\sqrt{a^2} = |a|$
$\sqrt{ab} = \sqrt a\sqrt b$ for nonnegative $a,b$

Rewrite move: compare

8^2

and

2^5

by writing

8^2 = (2^3)^2 = 2^6

Trap:

\sqrt{a+b}

does not split into

\sqrt a + \sqrt b

8. Factoring and Quadratic Logic

Quadratics usually reward pattern recognition more than formulas. First move everything to one side so the equation equals zero. Then look for a greatest common factor, a simple trinomial, or a difference of squares.

x^2 - 9 = (x-3)(x+3)

x^2 - 5x = x(x-5)

If a product equals zero, at least one factor must be zero. That is often the whole problem.

High-value habit: never divide both sides by a variable unless you know it cannot be zero. Factoring protects hidden roots.

9. Functions and Clean Substitution

Function questions are mostly substitution questions with notation wrapped around them. Replace every input with parentheses to avoid sign mistakes.

Original example: If

f(x)=x^2-2x+1

, then

f(-3)=(-3)^2-2(-3)+1=16

For composed functions, work inside out. If $g(f(2))$ is asked, find $f(2)$ first, then feed that value into $g$ .

10. Word Problems and Translation Rules

GRE algebra word problems are translation exercises. Define the variable, convert each sentence into a relationship, and only then solve.

Phrase	Algebra
5 less than x	$x-5$
Twice the sum of x and 4	$2(x+4)$
x is 30% greater than y	$x=1.3y$
Ratio of a to b is 3 to 5	$a=3k,\ b=5k$

Translation rule: write the relationship in symbols before touching the answer choices or mental arithmetic.

11. Quantitative Comparison Strategy

Quantitative Comparison is not about finding one answer. It is about proving whether the relationship is always fixed.

If one legal case makes Quantity A larger and another makes Quantity B larger, the answer is cannot be determined.
Test positives, negatives, fractions, zero, and boundary cases whenever those values are allowed.
Pay special attention when variables are described only as numbers or reals. That leaves many cases open.

Classic GRE pattern: if

x > 0

, comparing

x

with

x^2

needs at least two tests, such as

x=2

and

x=1/2

12. Mini Diagnostic

D1. If

3x-7=20

, what is

x

D2. If

2a+b=9

, what is

4a+2b+1

D3. Solve

|x-4|=6

D4. Factor

x^2-16

D5. Compare Quantity A:

x

and Quantity B:

x^2

given

0<x<1

13. Final Revision Sheet

Core Rules

$a(b+c)=ab+ac$
$(a+b)^2=a^2+2ab+b^2$
$a^2-b^2=(a-b)(a+b)$
$\sqrt{a^2}=|a|$

GRE Habits

Write restrictions first.
Target the requested expression.
Factor before expanding if zero is involved.
Test more than one legal case in QC.

Final reminder: on GRE algebra, the shortest correct method is usually the best method.

Solved Examples

Try each prompt mentally first, then open the explanation and check whether your method was shorter than the written solution.

Example 1: If 5x - 2 = 23, what is x?

Add 2 to both sides to get $5x=25$ . Divide by 5, so $x=5$ .

Example 2: If 3p + 4q = 18, what is 6p + 8q - 7?

$6p+8q$ is double $3p+4q$ , so it equals 36. Then $36-7=29$ .

Example 3: Solve the inequality -2x + 3 < 11.

Subtract 3 to get $-2x<8$ . Divide by -2 and reverse the sign: $x>-4$ .

Example 4: Solve |2x + 1| = 9.

Case 1: $2x+1=9$ , so $x=4$ . Case 2: $2x+1=-9$ , so $x=-5$ .

Example 5: Simplify (x^2 - 25)/(x + 5).

Factor the numerator: $x^2-25=(x-5)(x+5)$ . Cancel the common factor to get $x-5$ , with the restriction $x\ne -5$ .

Example 6: If x^2 = 7x, what are the possible values of x?

Move everything to one side: $x^2-7x=0$ . Factor: $x(x-7)=0$ . So $x=0$ or $x=7$ .

Example 7: If f(t) = t^2 + 3t, what is f(-2)?

Substitute carefully: $(-2)^2+3(-2)=4-6=-2$ . So $f(-2)=-2$ .

Example 8: Given x > 0, compare x and x^2.

If $x=2$ , then $x^2>x$ . If $x=1/2$ , then $x>x^2$ . Therefore the relationship cannot be determined.

GRE Algebra FAQs

What GRE algebra topics matter most in Quant?

Linear equations, inequalities, factoring, quadratics, exponents, roots, functions, word-problem translation, and Quantitative Comparison are the most reusable GRE algebra skills.

Are these GRE Algebra notes original?

Yes. This page uses original teaching copy, fresh examples, and independently written practice prompts rather than copied prep-book explanations.

What is the fastest way to improve GRE algebra?

Improve the decision layer first: write restrictions, target expressions directly, factor before dividing, and test multiple legal values in Quantitative Comparison.

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