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SAT Algebra: Linear Equations and Inequalities

Linear equations and inequalities are the highest-frequency topic on Digital SAT Math (roughly 7–9 questions per test). Four solve patterns, one Desmos shortcut, and the inequality-flip trap that costs students the most points.

9 分钟 TEKS ALG SAT Math

Linear equations and inequalities are the highest-frequency topic on Digital SAT Math — roughly 7–9 questions per test. Master the four solve patterns and you've banked easy points across both modules.

Why linear equations dominate Module 1

Of the four SAT Math domains, Algebra is the biggest single category (~35% of the test). Inside Algebra, linear equations and inequalities are the workhorse — College Board uses them to calibrate the adaptive routing because they reveal so cleanly whether a student has fluency or not.

The four solve patterns
  1. One-variable solve — find x given an equation
  2. Slope-intercept — read m and b from y = mx + b
  3. Parallel / perpendicular — match or negate slopes
  4. Inequalities — same rules as equations, but flip when multiplying by a negative

Pattern 1: Solve for x

Every "solve for x" reduces to: distribute, combine, isolate, divide. Watch the sign of the coefficient.

3(2x − 4) + 7 = 5x + 1 6x − 12 + 7 = 5x + 1 6x − 5 = 5x + 1 x = 6 Distribute first. Then collect x on one side.

Pattern 2: Slope-intercept form

b = y-intercept rise run y = mx + b m = rise/run, b = where line crosses y-axis
Read slope (m) from the steepness and y-intercept (b) from where the line crosses the vertical axis.
Find the slope of the line through (2, 5) and (6, 13). m = (13 − 5) / (6 − 2) m = 8 / 4 = 2 Slope formula: m = (y₂ − y₁) / (x₂ − x₁). Order doesn't matter as long as you're consistent.

Pattern 3: Parallel and perpendicular

  • Parallel lines have the same slope. Different y-intercepts.
  • Perpendicular lines have slopes that multiply to −1 (negative reciprocals).
Find a line perpendicular to y = (3/4)x + 2 through (4, 1). Perpendicular slope: m = −4/3 y − 1 = −4/3 (x − 4) y = −(4/3)x + 16/3 + 1 = −(4/3)x + 19/3 Use point-slope form when you have a point and a slope.

Pattern 4: Inequalities

Inequalities work just like equations — with one critical rule: flip the inequality sign whenever you multiply or divide both sides by a negative number.

−2x + 5 ≤ 11 −2x ≤ 6 x ≥ −3 Divided by −2, so ≤ flipped to ≥. This is the #1 mistake on SAT inequality questions.

Desmos shortcut

For any linear equation, type it directly into Desmos. To find where two lines intersect (a system), graph both and click the intersection point — Desmos shows the coordinates instantly. This converts a 2-minute algebra problem into a 15-second visual check.

Word problems → linear equations

Most SAT linear-equation questions arrive as word problems. The template:

1. Identify the unknown — name it x (or use a meaningful letter) 2. Translate the sentence into an equation 3. Solve algebraically 4. Re-read the question — am I answering what was asked? Step 4 catches the trap where x is found but the question wants x + 3.
A phone plan charges $40/month plus $0.10 per text. The bill was $58. How many texts? Let t = number of texts 40 + 0.10t = 58 0.10t = 18 t = 180 texts "Plus" → addition. "Per" → multiplication. Translation matters more than algebra.

Compound inequalities

A compound inequality has two bounds. Treat both ends together.

−5 < 2x + 1 ≤ 9 −6 < 2x ≤ 8 (subtract 1 throughout) −3 < x ≤ 4 (divide by 2) Whatever you do to the middle, do to both ends.

Common mistakes

  • Forgetting to flip the inequality when dividing by a negative
  • Confusing slope (m) with y-intercept (b) when reading y = mx + b
  • Using the reciprocal but forgetting the negative sign for perpendicular lines
  • Distributing only the first term across parentheses (e.g., 3(2x − 4) ≠ 6x − 4)
  • Solving for x but answering "x + 3" — always re-read the final question

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