Linear inequalities

A linear inequality looks like a linear equation with the equals sign replaced: 3x + 5 < 17, or 2x − y ≥ 4. The difference in meaning is large. A linear equation in one variable usually has a single solution, while a linear inequality has a whole range of them. Solving 3x + 5 < 17 gives x < 4, meaning every number below 4 satisfies it. Solutions are written in inequality notation, in interval notation, or graphed on a number line.

The algebra is nearly identical to solving equations — isolate the variable by adding, subtracting, multiplying, and dividing on both sides. The one rule that has no equation counterpart is that multiplying or dividing both sides by a negative number flips the inequality symbol. Starting from −2x > 6 and dividing by −2 gives x < −3, not x > −3. A quick check confirms it: x = −4 makes the original true, while x = 0 does not.

In two variables, a linear inequality describes a region of the coordinate plane rather than a line. Graph the boundary as if the inequality were an equation, then decide whether the line is solid or dashed and which side to shade. Use a solid line for ≤ or ≥ because points on the line satisfy the inequality, and a dashed line for < or >. To find the correct side, test a point off the line — (0, 0) is easiest when it works. A system of linear inequalities is solved by graphing each and taking the overlapping region.

Linear inequalities are tested across math sections. The SAT covers them directly in algebra and in systems of equations and inequalities, the AMC includes them in its algebra material, and the Praxis Core math test pairs solving inequalities with solving equations. Sign-flip errors and dashed-versus-solid boundary mistakes are the two most common ways points are lost.