Solve linear equations, two-variable systems, inequalities, and absolute-value statements — and read their solution sets on a number line — for the GRE Quant section.
$m$ is the slope, $b$ is the $y$-intercept; a system solves where two such lines meet.
Multiply or divide by a negative reverse the sign.
Adding or subtracting a negative never flips the sign.
Requires $k \ge 0$; always two cases.
For $>$ instead, split into two rays: $x - c < -k$ or $x - c > k$.
A linear equation relates quantities that change at a constant rate; graphed, it is a straight line, and algebraically it is the object you will manipulate more than any other on the GRE Quantitative section. The test rarely asks you to just solve . Instead it buries linear relationships inside systems, inequalities, absolute-value constraints, and Quantitative Comparison traps where the "obvious" answer is wrong. This guide builds the full toolkit — solving, systems, inequalities, and absolute value — with the GRE's favorite twists in view.
To isolate a variable, undo operations in reverse order: clear parentheses and fractions, collect the variable on one side and constants on the other, then divide by the coefficient.
Solve .
5x - 10 + 3 &= 2x + 10 \\ 5x - 7 &= 2x + 10 \\ 3x &= 17 \\ x &= \frac{17}{3} \end{aligned}$$ A non-integer answer is completely normal — the GRE does not reward "clean" numbers, and grid-in (Numeric Entry) questions frequently want a fraction or decimal. ## Systems of Two Equations A **system** is two equations sharing the same variables; the solution makes *both* true at once. Two methods cover everything. **Substitution** is best when one variable is already isolated. To solve $y = 2x + 1$ and $3x + y = 16$, replace $y$: $$3x + (2x + 1) = 16 \;\Rightarrow\; 5x + 1 = 16 \;\Rightarrow\; x = 3,\; y = 7.$$ **Elimination** shines when both equations are in $ax + by = c$ form. Scale one equation so a coefficient pair becomes opposite, then add. A GRE-favorite shortcut: when a question wants a *combination* like $x + y$, adding or subtracting the equations can deliver it in one step, with no need to find $x$ and $y$ separately. | Lines | Slopes / intercepts | Number of solutions | |---|---|---| | Cross once | Different slopes | Exactly one | | Same line | Same slope **and** intercept | Infinitely many | | Parallel | Same slope, different intercept | None | ## Linear Inequalities — The One Rule That Trips Everyone An **inequality** uses $<$, $>$, $\le$, or $\ge$ and describes a *range* of values. Solve it exactly like an equation with **one crucial exception**: when you multiply or divide both sides by a **negative** number, **reverse** the inequality sign. **Solve $-4x + 9 \ge 25$.** $$-4x \ge 16 \;\Rightarrow\; x \le -4.$$ The sign flipped because we divided by $-4$. Adding or subtracting a negative never flips the sign — only multiplying or dividing does. Confirm by testing a value from your range: $x = -5$ gives $-4(-5) + 9 = 29 \ge 25$. ✓ ## Absolute Value: Two Cases, Always The absolute value $\lvert a \rvert$ is the distance of $a$ from zero, so it is never negative. Two structures matter: - **Equation** $\lvert x - c \rvert = k$ (with $k \ge 0$) splits into $x - c = k$ **or** $x - c = -k$ — two solutions. - **Inequality** $\lvert x - c \rvert < k$ becomes the *sandwich* $-k < x - c < k$, a single interval. But $\lvert x - c \rvert > k$ becomes *two* rays: $x - c < -k$ **or** $x - c > k$. For example, $\lvert 2x - 5 \rvert = 9$ gives $2x - 5 = 9$ ($x = 7$) or $2x - 5 = -9$ ($x = -2$). And $\lvert x - 3 \rvert < 5$ means $-5 < x - 3 < 5$, i.e. $-2 < x < 8$ — every point within 5 units of 3, a segment you can shade on a number line. ## The Quantitative Comparison Angle Many GRE algebra items are Quantitative Comparisons: you compare Quantity A and Quantity B and decide whether A is greater, B is greater, they are equal, or the relationship **cannot be determined**. The trap is settling on a single value when a variable can range. If $\lvert x + 1 \rvert = 4$, then $x = 3$ *or* $x = -5$ — so comparing $x$ to $3$ yields "equal" in one case and "B greater" in another. The correct choice is *cannot be determined*. Whenever a constraint permits more than one value, test the extremes before committing. ## A Reliable Game Plan - **Solve for a variable?** Distribute, collect, divide — a fraction is fine. - **System given?** Isolated variable → substitute; standard form → eliminate; want $x \pm y$ → add/subtract directly. - **Inequality?** Same steps as an equation, but flip when you divide by a negative. - **Absolute value?** Split into two cases; for $<$ make one sandwich, for $>$ make two rays. - **Quantitative Comparison?** Probe multiple legal values before deciding — "cannot be determined" is a real answer.Solve for .
At a stand, a sandwich costs $3 more than a drink. Two sandwiches and three drinks cost $26. What is the price of one sandwich?
Solve the system and .
Solve and state the solution.
Find all solutions of , then report their sum.
Solve and describe the solution set on a number line.
Forgetting to flip the inequality sign after dividing by a negative — solving as .
Whenever you multiply or divide by a negative, reverse the sign: . Test a value to confirm the direction.
Solving only one case of an absolute-value equation, e.g. taking to give only .
Every statement splits into two equations, one with and one with : here or .
Treating an absolute-value inequality as a single equation, or writing as .
For "", sandwich into one interval; for "", split into two rays. means or , not a middle band.
On Quantitative Comparison, plugging in one convenient value and concluding a definite relationship.
Probe multiple legal values, especially negatives, fractions, and 0. If the comparison flips, the answer is "cannot be determined."
If , what is the value of ?
In the system and , what is the value of ?
Which of the following describes all solutions to ?
If , what is the sum of all values of that satisfy the equation?
It is given that .
Quantity A:
Quantity B:
The inequality is equivalent to which of the following?
Simplify and factor expressions, expand with FOIL, and translate GRE age, mixture, interest, and work stories into equations you can actually solve.
Factor, expand, and solve quadratics for GRE Quant — zero-product property, the quadratic formula, the discriminant, and reading a parabola.
Master function notation, slope, distance, midpoint, and the equation of a line for GRE Quant — plus reading values straight off a graph.