Linear Inequalities in One or Two Variables: SAT Practice Questions & Study Guide
Solving and graphing inequalities, interpreting solution sets, and modeling real-world constraints with inequality expressions.
Understanding Linear Inequalities in One or Two Variables on the SAT
Linear inequalities extend the concept of linear equations by replacing the equals sign with an inequality symbol (<, ≤, >, ≥). The solution is no longer a single point but a range of values (in one variable) or a half-plane (in two variables). On the Digital SAT, inequality questions appear as both pure algebra (solve for x and report the solution set) and applied problems (model a constraint and find which values or which region satisfies it).
Solving a linear inequality follows the same rules as solving an equation with one critical exception: when you multiply or divide both sides by a negative number, you must flip the inequality sign. For example, -2x < 8 becomes x > -4 after dividing both sides by -2. This rule trips up a large fraction of students on the SAT, so flag every step where you divide or multiply by a negative.
Compound inequalities (e.g., -3 < 2x + 1 ≤ 7) require performing the same operations on all three parts simultaneously. Solving -3 < 2x + 1 ≤ 7: subtract 1 from all parts to get -4 < 2x ≤ 6, then divide all by 2 to get -2 < x ≤ 3. The Digital SAT often asks for the 'greatest possible integer value' or 'minimum value' satisfying such a compound inequality.
For two-variable inequalities on the SAT, you usually need to identify which region of the plane satisfies the inequality, or determine which point satisfies a system of inequalities. The solid-line vs. dashed-line distinction (≤ vs. <) and the shading direction (above vs. below the line) are tested. A quick shortcut: substitute the test point (0, 0) into the inequality; if it satisfies, shade the side containing the origin; if not, shade the other side.
Key Rules & Formulas
Memorize these rules — they come up directly in SAT questions.
Flip the inequality sign when multiplying or dividing both sides by a negative number.
-3x > 12: divide both sides by -3 and flip: x < -4.
For compound inequalities, apply each operation to all three parts.
1 ≤ 2x - 3 < 9: add 3 to get 4 ≤ 2x < 12; divide by 2 to get 2 ≤ x < 6.
The solution set of a one-variable inequality is all values satisfying it; express with interval notation or a number line.
x > 5 means all real numbers greater than 5 (not including 5).
A ≤ or ≥ inequality uses a closed boundary (included); < or > uses an open boundary (excluded).
x ≥ 3 includes 3; x > 3 does not include 3.
To find the region satisfying ax + by ≤ c, test the point (0, 0): if it satisfies the inequality, shade toward the origin.
2x + y ≤ 6: test (0,0): 0 ≤ 6 ✓ → shade the side containing the origin.
Linear Inequalities in One or Two Variables Practice Questions
Select an answer and click Check Answer to reveal the full explanation. Questions go from easiest to hardest.
Which value of x satisfies the inequality 3x - 5 > 10?
Which of the following is the solution set of -2x + 4 ≤ 10?
A student needs to score at least 75 points on a test. She has already answered questions worth 52 points correctly. If she answers the remaining questions, each worth p points, and answers at least 3 more correctly, which inequality represents this situation?
What is the largest integer value of x that satisfies 4x - 3 < 2x + 9?
Which of the following values of x satisfies both x + 3 > 7 and 2x - 1 < 15?
The compound inequality -7 ≤ 3x + 2 < 14. What is the range of values for x?
A gym charges a $30 enrollment fee and $20 per month. If Jorge can spend at most $150 total, what is the maximum number of complete months he can be a member?
Which point (x, y) is a solution to the system of inequalities y < 2x + 1 and y ≥ -x + 5?
If -1 < (x - 3)/2 ≤ 4, what is the range of values for x?
In the xy-plane, the solution to the system y ≤ 3x - 1 and y > -2x + 4 contains the point (k, 2). What is the range of possible values for k?
Want more Linear Inequalities in One or Two Variables practice?
Access 1,000+ additional questions filtered by difficulty and score band in the full 1600.lol question bank — free, no signup needed.
Common Mistakes to Avoid
These are the most frequent errors students make on Linear Inequalities in One or Two Variables questions. Knowing them in advance prevents costly point losses.
- !Forgetting to flip the inequality sign when dividing or multiplying by a negative number.
- !On compound inequalities, performing the operation only on the middle part and not on both boundaries.
- !Confusing ≤ with < when reporting the solution set, causing inclusion/exclusion errors.
- !When a word problem says 'at most' or 'at least,' translating these into the wrong inequality direction ('at most 10' means ≤ 10, not ≥ 10).
- !For two-variable systems, shading the wrong half-plane by not testing a specific point to confirm the direction.
SAT Strategy Tips: Linear Inequalities in One or Two Variables
Highlight or underline inequality symbols as you read—it is easy to misread ≤ as < under time pressure, which changes the answer.
When in doubt about which region satisfies a two-variable inequality, test (0, 0) in the inequality—it takes five seconds and removes the guesswork.
For 'maximum integer' or 'minimum integer' questions, solve the inequality first, then identify the relevant extreme integer within the solution set.
Backsolving works especially well for one-variable inequality questions with numerical answer choices—just plug each choice into the inequality.
Other Algebra Subtopics
Linear Equations in One Variable
Solving equations with a single unknown, from simple one-step problems to multi-step equations with fractions and parentheses.
Linear Equations in Two Variables
Interpreting and writing equations relating two quantities, including converting between equation forms and identifying key features like slope and intercepts.
Linear Functions
Understanding linear functions as rules mapping inputs to outputs, evaluating them, and interpreting function notation in real-world models.
Systems of Two Linear Equations
Finding the intersection of two lines by substitution or elimination, and analyzing when systems have no solution or infinitely many solutions.
Master Linear Inequalities in One or Two Variables on the SAT
These 10 questions are just the start. Unlock the full 1600.lol question bank for 12,000+ official-style SAT questions with the Desmos calculator, instant feedback, and progress tracking.
Join 50,000+ students preparing for the 2025–2026 Digital SAT on 1600.lol