Write, solve and interpret linear inequalities and systems of inequalities, including their graphs. It is part of Algebra in the Math section. The book teaches it in Chapter 19, Linear Inequalities, with 12 practice questions, and tests it 4 more times across the two full-length practice tests.
About Chapter 19
A linear inequality compares two linear expressions with one of the symbols <, >, ≤ or ≥. Its solution is usually a whole range of values rather than a single number. The digital SAT asks you to solve one-variable inequalities, to write inequalities from real limits such as budgets and minimums, and to decide which points satisfy a two-variable inequality or a system of inequalities. These questions belong to the Algebra domain and can appear in either Math module.
What the question looks like
Solve; Typical stem: Which inequality represents all solutions to the given inequality?; What you produce: a simplified inequality such as x ≥ −3
Model; Typical stem: Which inequality represents this situation?; What you produce: an inequality or a system built from the story
Test points; Typical stem: Which point (x, y) is a solution to the given system of inequalities?; What you produce: the one point that satisfies every inequality
Interpret; Typical stem: What is the maximum number of items that can be bought?; What you produce: a whole number at the edge of the solution set
Read a graph; Typical stem: Which system of inequalities is represented by the graph?; What you produce: inequalities that match the boundary lines and the solution region
How to recognize it
The symbols are the obvious signal. In word problems, the signal is a phrase that sets a limit. Translate it before you do anything else.
at least, no less than, a minimum of; Symbol: ≥; Example: at least 12 people: p ≥ 12
at most, no more than, a maximum of, cannot exceed; Symbol: ≤; Example: at most $40: c ≤ 40
more than, greater than, exceeds; Symbol: >; Example: more than 3 hours: h > 3
fewer than, less than, below; Symbol: <; Example: fewer than 30 seats: s < 30
Trap: "No more than 50" sets an upper limit (≤ 50), and "no fewer than 50" sets a lower limit (≥ 50). Read the whole phrase, not just the word "more" or "fewer."
Common traps
Forgetting to flip after multiplying or dividing by a negative: Circle any negative coefficient before you divide, then test a value.
Flipping when you only added or subtracted: Only multiplying or dividing by a negative flips the symbol.
Reversing "at least" and "at most": "At least" is a floor (≥); "at most" is a ceiling (≤).
Rounding a whole-number answer to the nearest integer: Maximums round down; minimums round up. Check the neighboring value.
Counting a point on a dashed line: Substitute: if the two sides are equal and the symbol is < or >, the point is excluded.
Testing only one inequality of a system: A solution must make every inequality true.
Choosing the wrong side when y has a negative coefficient: Solve for y (flipping if you divide by a negative) or test (0, 0).
Video lessons
Two short lessons from independent YouTube creators, chosen for this skill. The notes are ours; the videos belong to their creators, who are not affiliated with this site.
Graphing linear inequalities in two variables (GoTutor Math, 10:11). Watch for: Solid or dashed boundary lines, and which side to shade.
Linear equations and inequalities (Brain Buffs Tutoring, 9:56). Watch for: Writing an inequality from a word problem and flipping the sign when you divide by a negative.
Practice this skill
The full method, worked examples and 12 practice questions with every answer choice explained are in Chapter 19 of the book and in the study workspace, where an AI coach can talk you through each one.
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