# Linear inequalities on the SAT Canonical: https://duckyhelper.com/learn/sat-math/linear-inequalities/ Updated: 2026-10-01 Solve a linear inequality the same way you solve an equation, with one extra rule: when you multiply or divide both sides by a negative number, flip the inequality sign. On the SAT you also turn word problems into inequalities ("at least" means \(\ge\), "no more than" means \(\le\)) and check which points satisfy a system of inequalities by plugging them in. ## The key idea An inequality describes a range of answers, not one number. Adding, subtracting, or multiplying by a positive number keeps the sign. Multiplying or dividing by a negative number reverses the order of numbers, so the sign must flip: $$ -2x < 6 \quad\Longrightarrow\quad x > -3 $$ **Words to symbols** | The question says | Symbol | Example | | --- | --- | --- | | at least, no less than, minimum | \(\ge\) | at least 8 notebooks: \(n \ge 8\) | | at most, no more than, maximum | \(\le\) | at most 2,000 pounds: \(w \le 2000\) | | more than, exceeds | \(>\) | more than $50: \(c > 50\) | | less than, fewer than | \(<\) | fewer than 30 people: \(p < 30\) | In the xy-plane, \(y > mx + b\) is the region above the line and \(y < mx + b\) is below it. A strict sign (\(<\) or \(>\)) uses a dashed line; \(\le\) or \(\ge\) uses a solid line because points on the line count. ## Worked examples **Example 1: dividing by a negative** Problem: Solve \(5 - 3x \le 17\). 1. Subtract 5 from both sides. $$ -3x \le 12 $$ 2. Divide by \(-3\). Dividing by a negative flips the sign. $$ x \ge -4 $$ 3. Check a value that should work, \(x = 0\): \(5 \le 17\) is true. Check one that should not, \(x = -5\): \(20 \le 17\) is false. Good. Answer: \(x \ge -4\) **Example 2: a word problem where you round down** Problem: A delivery van can carry at most 2,000 pounds. The driver weighs 180 pounds, and each box weighs 40 pounds. What is the greatest number of boxes the van can carry with the driver? 1. Let b be the number of boxes. Total weight is at most 2,000. $$ 180 + 40b \le 2000 $$ 2. Subtract 180. $$ 40b \le 1820 $$ 3. Divide by 40. $$ b \le 45.5 $$ 4. Boxes come in whole numbers, and 46 would be too heavy. Round down to 45. Check: \(180 + 40(45) = 1980\), which is under 2,000. Answer: 45 boxes **Example 3: a point in a system of inequalities** Problem: The point \((3, b)\) satisfies both \(y \ge x + 2\) and \(y < -2x + 12\). If b is an integer, what is b? 1. Put \(x = 3\), \(y = b\) into the first inequality. $$ b \ge 3 + 2 $$ 2. Put the same point into the second inequality. $$ b < -2(3) + 12 $$ 3. So \(b \ge 5\) and \(b < 6\). The only integer in that range is 5. Answer: \(b = 5\) **Example 4 (SAT-hard): two conditions in one context** Problem: Jada has $60 to spend on notebooks that cost $4 each and pens that cost $1.50 each. She needs at least 8 notebooks. What is the greatest number of pens she can buy? 1. Write both conditions. Let n be notebooks and p be pens. $$ 4n + 1.5p \le 60 $$ 2. Pens are greatest when she buys the fewest notebooks she is allowed, so \(n = 8\). $$ 32 + 1.5p \le 60 $$ 3. Subtract 32. $$ 1.5p \le 28 $$ 4. Divide by 1.5. That is about 18.67. $$ p \le \frac{56}{3} $$ 5. Round down: 19 pens would cost \(32 + 28.50 = 60.50\), which is too much. 18 pens cost \(32 + 27 = 59\). Answer: 18 pens ## Common mistakes - **Forgetting to flip.** \(-3x \le 12\) becomes \(x \ge -4\), not \(x \le -4\). Fix: circle any negative number you divide or multiply by. - **Flipping when you should not.** Subtracting a negative or dividing by a positive does not flip the sign. Fix: only multiply or divide by a negative causes a flip. - **Rounding the wrong way in context.** "At most" problems round down, because rounding up breaks the limit. "At least" problems round up. Fix: test your rounded number in the original inequality. - **Mixing up "more than" and "at least".** "More than 5" is \(> 5\); 5 itself does not count. "At least 5" is \(\ge 5\). - **Testing a point in only one inequality.** For a system, the point must satisfy every inequality. Fix: check each one separately. ## Quick methods > **Tip: Plug in the choices** > > For "which point is a solution" questions, substituting each choice is a real method and often the fastest. One false inequality rules a choice out. > **Note: Desmos shades regions** > > Desmos graphs inequalities like \(y < -2x + 12\) as shaded regions, with dashed or solid lines. Where the shadings overlap is the solution set of the system. It helps for picturing, but plugging in points is usually quicker for multiple choice. ## Practice **5 SAT-style questions** 1. Which describes all solutions to \(-4x + 7 > 23\)? A. \(x < -4\) B. \(x > -4\) C. \(x < 4\) D. \(x > -7.5\) Answer: \(x < -4\). Subtract 7: \(-4x > 16\). Divide by \(-4\) and flip: \(x < -4\). \(x > -4\) forgets the flip. \(-7.5\) comes from adding 7 instead of subtracting it. 2. A gym charges $25 to join and $18 per month. Maria wants to spend no more than $205 in total. Which inequality gives the number of months m she can pay for? A. \(25 + 18m \le 205\) B. \(25m + 18 \le 205\) C. \(25 + 18m \ge 205\) D. \(18 + 25m \ge 205\) Answer: \(25 + 18m \le 205\). The $25 is paid once and $18 is paid each month, so the cost is \(25 + 18m\). "No more than" means \(\le\). The other choices swap the fee and the rate or flip the sign. 3. Which point is a solution to the system \(y < 3x + 1\) and \(y \ge -x + 5\)? A. \((0, 6)\) B. \((2, 4)\) C. \((1, 5)\) D. \((3, 1)\) Answer: \((2, 4)\). For \((2, 4)\): \(4 < 7\) is true and \(4 \ge 3\) is true. \((0, 6)\) and \((1, 5)\) fail the first inequality, and \((3, 1)\) fails the second. 4. All solutions to \(2(x - 3) \ge 5x + 9\) can be written as \(x \le c\). What is c? A. \(-5\) B. \(-1\) C. \(1\) D. \(5\) Answer: \(-5\). Distribute: \(2x - 6 \ge 5x + 9\). Subtract \(5x\) and add 6: \(-3x \ge 15\). Divide by \(-3\) and flip: \(x \le -5\). 5. Student-produced response: an elevator can carry at most 1,500 pounds. A 210-pound worker rides with boxes that weigh 55 pounds each. What is the greatest number of boxes that can ride with the worker? Answer: 23. \(210 + 55b \le 1500\) gives \(55b \le 1290\), so \(b \le 23.45\). Round down to 23. Check: 24 boxes would make 1,530 pounds. ## Frequently asked questions ### Why do you flip the sign when dividing by a negative? Multiplying by a negative reverses the order of numbers. For example, \(2 < 5\), but \(-2 > -5\). So if you divide both sides of an inequality by a negative, the bigger side becomes the smaller side, and the sign has to turn around to stay true. ### What is the difference between a dashed and a solid line? A dashed line means points on the line are not solutions, which matches \(<\) or \(>\). A solid line means points on the line count, which matches \(\le\) or \(\ge\). The shading shows every other point that works. ### How do I know whether to round up or down? Ask what the limit protects. A weight or budget limit ("at most") means you round down, or you go over. A minimum ("at least 120 points") means you round up, or you fall short. Always test the rounded number. ## Sources - [College Board: SAT Math, Algebra skills (linear inequalities in 1 or 2 variables)](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/algebra), accessed 2026-10-01 ## Related - [Linear equations in one variable on the SAT](https://duckyhelper.com/learn/sat-math/linear-equations/) - [Systems of linear equations on the SAT](https://duckyhelper.com/learn/sat-math/systems-of-equations/) - [How to solve and graph inequalities](https://duckyhelper.com/learn/algebra-1/inequalities/) - [SAT Math study guides](https://duckyhelper.com/learn/sat-math/) ## Try asking Ducky - "When exactly do I flip the inequality sign?" - "I got 46 boxes. Why is the answer 45?" - "Can you turn this word problem into an inequality with me?" ## Get DuckyHelper Free to start. The web app works in any browser, Chromebooks included; the Mac app can also draw on your real screen. 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