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Linear Equations
& Inequalities

Master the foundations of solving, graphing, and interpreting linear relationships: the most tested algebra skill on the GRE.

GRE Quantitative

What Is a Linear Equation?

A linear equation is any equation that can be written in the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants and \( x \) is a variable raised to the first power. Its graph is always a straight line.

On the GRE, you will solve for unknown values, compare quantities using linear relationships, and interpret graphs: all built on this single idea.

Visualizing \( y = 2x + 1 \)

1 2 3 0 1 2 3 4 5 x y (0, 1) (1, 3) (2, 5) run = 1 rise = 2

What Is a Linear Inequality?

A linear inequality replaces the equals sign with one of four inequality symbols. Instead of one solution, an inequality describes a range of solutions.

\( x > 3 \)
Strict greater than: excludes 3
\( x \geq 3 \)
Greater than or equal: includes 3
\( x < 3 \)
Strict less than: excludes 3
\( x \leq 3 \)
Less than or equal: includes 3

Why This Matters on the GRE

Quantitative Comparison
Compare two algebraic expressions: you must manipulate linear equations and reason about inequality direction.
Word Problems
Translate sentences into \( ax + b = c \) or \( ax + b < c \), then solve: the most common question type.
Data Interpretation
Read a graph and extract slope, intercept, or threshold values from a linear trend.

Linear equations and inequalities appear in roughly 25-30% of GRE Quantitative questions, either directly or as a step within a larger problem.

Core Rules: Solving Equations

1
Addition/Subtraction Property: Add or subtract the same value from both sides without changing the solution.
2
Multiplication/Division Property: Multiply or divide both sides by the same nonzero value without changing the solution.
3
Distributive Property: Use \( a(b + c) = ab + ac \) to clear parentheses before combining like terms.

Core Rules: Solving Inequalities

Inequalities follow the same addition, subtraction, multiplication, and division rules as equations, with one critical exception:

\( -2x > 6 \) \( \div (-2) \) \( x < -3 \) Sign flips! \( > \) becomes \( < \)

This rule is the single most tested inequality concept on the GRE. Forget to flip, and your answer will be wrong every time.

Equations vs. Inequalities

Equation Inequality
Symbol \( = \) \( <,\; >,\; \leq,\; \geq \)
Solution Type Single value (point) Range of values (interval)
Graph (1-D) A dot on the number line A ray or segment on the number line
Graph (2-D) A line A shaded half-plane
Multiply by negative No effect on \( = \) Flips the sign

How to Solve Step by Step

1
Distribute to clear any parentheses.
2
Combine like terms on each side separately.
3
Move variable terms to one side and constants to the other using addition/subtraction.
4
Isolate the variable by dividing (or multiplying). If dividing by a negative in an inequality, flip the sign.
5
Check your answer by substituting back into the original expression.

Example 1: Solve an Equation

1
Distribute: \( 3x - 12 + 2 = 5x - 18 \)
2
Combine: \( 3x - 10 = 5x - 18 \)
3
Move variables: Subtract \( 3x \): \( -10 = 2x - 18 \)
4
Move constants: Add 18: \( 8 = 2x \)
5
Isolate: Divide by 2: \( x = 4 \)

Example 2: Solve an Inequality

1
Subtract 3: \( -5x \leq 25 \)
2
Divide by \( -5 \): \( x \geq -5 \) (sign flips!)
−7 −6 −5 −4 −3 −2 −1 0 \( x \geq -5 \)

Example 3: System of Equations

1
Add the equations (elimination): \( 2x + y + x - y = 7 + 2 \) gives \( 3x = 9 \)
2
Solve for \( x \): \( x = 3 \)
3
Substitute back: \( 3 - y = 2 \), so \( y = 1 \)
1 2 3 4 1 2 3 x y \( 2x + y = 7 \) \( x - y = 2 \) (3, 1)

Compound Inequalities

A compound inequality combines two inequalities into one statement. On the GRE, you will encounter two types:

AND (Intersection)
\( -2 < x \leq 5 \): \( x \) must satisfy both conditions simultaneously. Solution is the overlap.
OR (Union)
\( x < -1 \) or \( x > 4 \): \( x \) satisfies at least one condition. Solution is everything outside the gap.

Absolute Value and Linear Inequalities

The absolute value \( |ax + b| \) measures distance from zero. It creates two cases:

\( |ax + b| = c \)
Split into: \( ax + b = c \) or \( ax + b = -c \). Produces two solutions.
\( |ax + b| < c \)
Becomes: \( -c < ax + b < c \). A "sandwich" compound inequality (AND).
\( |ax + b| > c \)
Becomes: \( ax + b > c \) or \( ax + b < -c \). An OR compound inequality.

Common Mistakes

✕ Forgetting to Flip
Dividing both sides of an inequality by a negative number without reversing the sign. This is the #1 error.
✕ Distributing Incorrectly
Writing \( 3(x - 4) = 3x - 4 \) instead of \( 3x - 12 \). Always multiply the coefficient by every term inside.
✕ Open vs. Closed Circles
Using a filled circle for strict inequalities (\( < \) or \( > \)) or an open circle for \( \leq \) / \( \geq \). Filled = included.

Summary Reference Table

Operation Equation Effect Inequality Effect
Add/subtract constant Solution unchanged Direction unchanged
Multiply/divide by positive Solution unchanged Direction unchanged
Multiply/divide by negative Solution unchanged Direction reverses
Combine like terms Always valid Always valid
Substitute to check Must get exact \( = \) Must satisfy direction

Question 1 of 5

Solve for \( x \): \( 4x - 7 = 2x + 9 \)

Question 2 of 5

Solve: \( -3x + 6 \geq 15 \)

Question 3 of 5

If \( 2x + 3y = 12 \) and \( x = 3 \), what is the value of \( y \)?

Question 4 of 5

Which value of \( x \) satisfies \( |2x - 1| < 5 \)?

Question 5 of 5

A line has slope \( m = -\frac{2}{3} \) and passes through the point \( (6, 1) \). What is the y-intercept?