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SAT Math Unit 1: Linear equations & inequalities in one variable

Most SAT algebra questions come down to setting up a linear equation and solving it cleanly. Watch for questions that ask about the number of solutions instead of the solution itself.

Covers: Solving linear equations, no-solution and infinite-solution cases, word problems, linear inequalities

Study guide

A linear equation in one variable has the variable only to the first power, like 3x + 5 = 20. To solve it, undo operations in reverse order: distribute and combine like terms on each side, move the variable terms to one side and the constants to the other, then divide by the coefficient. If fractions appear, multiply every term by the common denominator first so you work with whole numbers.

Some equations have no solution or infinitely many. When the x terms cancel and you're left with a false statement (like 4 = 9), there is no solution. When you're left with a true statement (like 4 = 4), every value of x works. The SAT often asks for the value of a constant that creates one of these cases: match the x coefficients, then make the constants different (no solution) or equal (infinitely many).

Word problems ask you to translate: "per" means a rate, "more than" means add, and "is" means equals. Define the variable, write the equation, then check that your answer makes sense in context. Linear inequalities are solved like equations, except that multiplying or dividing by a negative number flips the inequality sign. Inequality answers are ranges, so word problems often ask for the greatest or least whole number that works.

How the Math section is built: 44 questions in two modules of 22 (two in each module are unscored trial questions), 35 minutes per module, about 1.6 minutes per question. Roughly 35% of questions are Algebra, 35% Advanced Math, 15% Problem-Solving and Data Analysis, and 15% Geometry and Trigonometry, and about 30% are word problems in context. The second module gets harder or easier based on the first, and its hard questions decide top scores. About a quarter of the questions are student-produced responses, where you type your own answer instead of picking one.

Rules for typed answers: a positive answer can use up to 5 characters and a negative answer up to 6 (counting the minus sign). Fractions and decimals both work; if a decimal is too long, round or cut it off to fill the space (for 2/3, enter 2/3, .6666 or .6667, but not .66). Write mixed numbers as an improper fraction or decimal (7/2 or 3.5, never 3 1/2), and don't type units, percent signs, commas or dollar signs. If several answers are correct, enter just one.

Tools you get: a built-in Desmos graphing calculator for the whole section and a reference sheet. Use Desmos to graph both sides of an equation and read the intersection, find a vertex or zeros by tapping the graph, add a slider for 'for what value of k' questions, and run a regression on table data. Watch for questions that ask for an expression like 2x or x + y instead of x, and convert Desmos decimals into an allowed answer form. The reference sheet gives area and volume formulas, the Pythagorean theorem, special right triangle ratios, 360° (2π radians) in a circle and 180° in a triangle. It does NOT give slope, the circle equation, the vertex formula −b/2a, the discriminant, exponent rules, exponential growth forms, arc length and sector area, sine-cosine complement rules or parallel-line angle rules, so learn those.

To solve an equation for one variable in terms of the others, treat every other letter as if it were a number and use the same inverse operations. For example, to solve 3a + 2b = c for b, subtract 3a and then divide by 2: b = (c − 3a)/2. An absolute value equation |x − a| = b with b > 0 splits into two linear equations, x − a = b and x − a = −b, so it usually has two solutions. If b = 0 there is one solution, and if b < 0 there is none, because an absolute value is never negative.

A compound inequality like 2 < 3x − 1 ≤ 11 is solved by doing the same operation to all three parts at once, giving 1 < x ≤ 4. To test values against a compound inequality, check that each value satisfies both parts; "and" means both conditions hold, while "or" means at least one does.

Key ideas

• Isolate the variable with inverse operations; clear fractions early by multiplying through.
• ax + b = ax + c has no solution when b ≠ c and infinitely many when b = c.
• Translate words into an equation: "is" means =, "per" means divide, "more than" means add.
• Flip an inequality when you multiply or divide by a negative.
• Answer what's asked. If they want 2x + 1, you may not need x.

Formulas and rules

Common mistakes

• Solving for x when the question asks for an expression like 3x − 2.
• Forgetting to distribute a negative sign across parentheses.

Words to know

No-solution linear equation

ax + b = ax + c with b ≠ c: the x terms cancel and leave a false statement.

Infinitely many solutions (one equation)

Both sides are identical (b = c), so every x works.

Clearing fractions

Multiply every term by the common denominator.

"Per" (word problems)

A rate: divide one quantity by the other.

Inequality flip rule

Multiplying or dividing by a negative reverses the inequality sign.

Linear equation

An equation where the variable is only to the first power, like 3x + 5 = 20.

Inverse operations

Operations that undo each other (add/subtract, multiply/divide), used to isolate a variable.

Linear inequality

An inequality like 2x − 3 < 7 whose solution is a range of values.

Translating "more than"

"5 more than x" means x + 5.

Defining a variable

Stating what the letter stands for, with units, before writing an equation.

Sample practice questions

157 more questions like these, from level 1 to 5, are in the practice set.