# Linear Inequalities on the SAT: Examples & Practice | 1600.now Source: https://1600.now/sat-skill/linear-inequalities --- Math · Algebra ## Linear Inequalities Solve inequalities like equations, but reverse the comparison when multiplying or dividing by a negative number. A solution must also respect whether the boundary is included. Written by [Luke Finigan](https://1600.now/about) 2 min read Updated Oct 2, 2026 ### Reverse the comparison only for negative multiplication or division For $7-3x>16$, subtract 7 to get $-3x>9$. Dividing by $-3$ reverses the sign: $x<-3$. Test $x=-4$: the original left side is 19, which is greater than 16. Moving a term by adding or subtracting does not itself reverse an inequality. The reversal comes from multiplying or dividing both sides by a negative quantity. ### Translate limits into inclusive or strict boundaries A budget of at most 50 dollars gives a $\le$ constraint. A height greater than 120 centimeters gives a $>$ constraint. Equality is allowed in the first case and excluded in the second. If notebooks cost 4 dollars each and shipping is 6 dollars, a 30-dollar budget gives $4n+6\le30$, hence $n\le6$. Since $n$ counts notebooks, use nonnegative whole numbers, not every real value below 6. ### Test points in two-variable inequalities For $y<2x+1$, draw the boundary $y=2x+1$ as a dashed line because equality is excluded. The point $(0,0)$ works since $0<1$, so the side containing that point is shaded. A system requires the overlap. A point satisfying only one inequality is not feasible. Substitution is a direct check when the answer choices are ordered pairs. ### Check maximum and minimum requests If $2p+3q\le18$ with nonnegative integers $p$ and $q$, the maximum $q$ occurs at $p=0$, giving $q=6$. Adding an extra condition such as $p\ge2$ changes the bound to $q\le\frac{14}{3}$, so the largest integer is 4. A numerical boundary is not always an allowed answer. Apply the integer restriction and strictness after the algebra, then test the proposed endpoint. ### Check your understanding Try it yourself: Which inequality is equivalent to $-2x\le8$? · A $x\le-4$ B $x\ge-4$ C $x\le4$ D $x\ge4$ ### Sample linear inequalities questions Linear Inequalities · Easy For a certain category, the positive mass $m$, in grams, of an object must be less than 640 grams. Which inequality represents this situation? A $m \gt 640$ B $m \lt 0$ C $-640 \lt m \lt 0$ D $0 \lt m \lt 640$ - [Open the question and explanation](https://1600.now/bank/math/7404b5ce) ### Practice this skill 1. Translate each phrase into a comparison symbol before solving. 2. Test a boundary and one interior value in the original inequality. 3. For two-variable systems, check each constraint separately. [Practice linear inequalities](https://1600.now/bank/math/skill/Linear%20inequalities%20in%20one%20or%20two%20variables) [Print the linear inequalities worksheet and worked answers](https://1600.now/sat-linear-inequalities-worksheet). Then use a [mixed practice module](https://1600.now/modules) to check whether you can choose this method among other question types. ### Source and question labels The bank label for this guide is "Linear inequalities in one or two variables" in Algebra. The worked examples above are written for this guide. - [College Board: Types of Math tested](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types) ### Related skills - **[Linear Equations in One Variable](https://1600.now/sat-skill/linear-equations-one-variable)** Keep an equation balanced: simplify each side, collect the variable terms, and undo multiplication or division. - **[Linear Equations in Two Variables](https://1600.now/sat-skill/linear-equations-two-variables)** A two-variable linear equation describes a line. - **[Systems of Linear Equations](https://1600.now/sat-skill/systems-of-linear-equations)** A solution to a system must satisfy both equations.