GRE Algebra Notes, Formulas and Solved Examples

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.

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

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

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.

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

Two equations may determine two variables, but they can also describe the same line or contradict each other. Elimination removes a variable by adding suitable multiples of the equations; substitution replaces a variable with an equivalent expression.

If $2r+s=13$ and $r-s=2$, adding gives $3r=15$, so $r=5$ and $s=3$. Check both original equations.

By contrast, $x+y=4$ and $2x+2y=8$ give infinitely many real pairs. Replacing 8 with 9 makes the equations inconsistent. Count independent conditions before claiming a unique answer.

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.

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

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.

For the product above, the complete solution is x < −5 or x > 2. Neither endpoint is included. For a rational inequality, also mark excluded denominator zeros. Never multiply by a variable expression without knowing its sign.

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

Conditions: the table assumes a > 0. If a = 0, |x| = a has only x = 0, |x| < a has no solution, and |x| > a allows every nonzero real x. If a < 0, equality and the less-than inequality have no solutions; the greater-than inequality holds for every real x.

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 $a,b$

Conditions: the negative-power rule requires a nonzero base. For integer exponents, keep every original expression defined; 0⁰ and division by zero are not permitted. Fractional exponents need extra care over the reals, so do not apply positive-base rules blindly to negative bases.

Squaring can introduce extra solutions because a negative and a positive number have the same square. First record the domain and the required sign of the side equal to a principal square root, then square and check every candidate in the original equation.

For $\sqrt{x+6}=x$, the right side requires $x\geq0$. Squaring gives $x+6=x^2$, or $(x-3)(x+2)=0$. The candidate $-2$ is excluded. At $x=3$, both original sides equal 3, so the only solution is 3.

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.

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

If a real quadratic $ax^2+bx+c=0$ does not factor conveniently, with $a\ne0$, use:

The discriminant $D=b^2-4ac$ describes real roots: D > 0 gives two distinct real roots; D = 0 gives one repeated real root; D < 0 gives no real roots. For $x^2+x-1=0$, D = 5 and the roots are $(-1\pm\sqrt5)/2$.

The sum of the roots is $-b/a$ and their product is $c/a$ when roots are counted with multiplicity. If a problem requires real solutions, check D before using those relationships to claim such solutions exist.

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

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

For distinct points with different x-coordinates, slope is $m=(y_2-y_1)/(x_2-x_1)$. A nonvertical line can be written $y=mx+b$, where b is its y-intercept. Substitute a known point after calculating m.

The line through (2, 1) and (6, 9) has slope $(9-1)/(6-2)=2$. Substituting (2, 1) gives $1=4+b$, so the equation is $y=2x-3$. Set y = 0 to get the x-intercept 3/2.

A horizontal line has slope 0. A vertical line x = c has undefined slope and cannot be written as y = mx + b. For two nonvertical lines, parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

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

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.

A changing relationship can also mean equality in one case and a strict inequality in another. Several examples with the same outcome do not prove that outcome for every legal input; use an algebraic argument when the relationship is fixed.