Inequalities
Lessons in this chapter
About Inequalities
Inequalities is a short but essential extension of Linear Equations, replacing the equals sign with <, >, ≤, or ≥ and asking kids to solve for a range of values instead of one.
Solve linear inequalities uses the same inverse-operation moves from Linear Equations — undoing addition, subtraction, multiplication, or division on both sides — to isolate the variable, but the answer this time is a range like x > 5 rather than a single value like x = 5.
The one genuinely new rule this chapter introduces is what happens when multiplying or dividing both sides by a negative number: the inequality sign flips direction, so x/(−2) < 4 solves to x > −8, not x < −8 — a step that has no equivalent in solving equations and is worth double-checking every time a negative appears.
Frequently asked questions
A linear inequality compares two expressions using <, >, ≤, or ≥ instead of an equals sign — x + 3 < 10 is an inequality, x + 3 = 10 is an equation. The equation has exactly one solution; the inequality is true for a whole range of values.
Isolate the variable using the same inverse operations as solving an equation — add, subtract, multiply, or divide both sides by the same amount. Solve Linear Inequalities works through both one-step cases (like x − 4 > 9) and multi-step cases that need simplifying first, the same progression Linear Equations followed.
Multiplying or dividing by a negative number reverses the order of every number on the number line — if 2 < 5, multiplying both sides by −1 gives −2 and −5, and −2 is actually greater than −5, not less. Flipping the inequality sign is what keeps the statement true after that reversal.
Solving correctly but forgetting to flip the sign — for example, solving −2x > 8 by dividing both sides by −2 and writing x > −4 instead of the correct x < −4. This is the single most common error in this chapter, precisely because it has no equivalent step in solving equations.
An equation’s answer is one specific value, like x = 8. An inequality’s answer is every value that makes the statement true — x > 5 includes 5.001, 6, 100, and every other number greater than 5, not just one of them.
As a ray on a number line: an open circle at the boundary number if it’s not included (< or >), a closed circle if it is (≤ or ≥), with shading extending in the direction of every value that satisfies the inequality.