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An inequality uses symbols (<, >, <=, >=) to show that one expression is less than, greater than, or not equal to another. Solving a one-step inequality uses the same inverse operations as solving a one-step equation, with one critical exception: when you multiply or divide both sides by a negative number, you must reverse the inequality sign. The solution to an inequality is not a single number but a set of numbers, represented on a number line with a ray. For example, x + 3 > 7 gives x > 4, meaning every number greater than 4 is a solution.
Start by connecting to equations — solve x + 3 = 7 to get x = 4, then ask "what if x + 3 needs to be greater than 7?" Test specific values to verify the solution. Teach number line graphing with open vs. closed circles (strict vs. inclusive). Introduce the sign-flip rule with a concrete example: if 2 < 5, then −2 > −5 (multiplying by −1 reverses the order). Test the sign-flip with substitution.
You already know how to solve one-step equations like x + 3 = 7 by applying an inverse operation to both sides: subtract 3 from both sides to get x = 4. Inequalities work almost exactly the same way — the only difference is that instead of one precise answer, you get an entire set of answers, and you use inequality symbols (< less than, > greater than, ≤ less than or equal, ≥ greater than or equal) to describe which values qualify. For x + 3 > 7, subtract 3 from both sides to get x > 4. Every number greater than 4 is a solution — not just 5, but also 4.1, 100, or 1,000,000.
The number line becomes your new best tool for displaying these solution sets. You learned on the integers-and-number-line that numbers increase to the right and decrease to the left. For x > 4, draw a open circle at 4 (to show 4 itself is not included) and shade all numbers to the right. For x ≥ 4, use a closed (filled) circle to show 4 is included. The open/closed circle distinction corresponds directly to the strict vs. inclusive inequality symbols: < and > exclude the endpoint, ≤ and ≥ include it. If you're ever unsure about the direction of shading, substitute a test value — pick any number from the shaded side and check that it satisfies the original inequality.
There is one critical rule that has no counterpart in equations: when you multiply or divide both sides by a negative number, the inequality sign flips direction. Here's the geometric reason. On the number line, multiplying by −1 reflects every point across zero: 2 maps to −2, 5 maps to −5. This reflection reverses the ordering of all numbers. Since 2 < 5 on the original line, after the reflection −2 > −5. Whenever you apply this reflection (by multiplying or dividing by a negative), all the "greater than" relationships become "less than" and vice versa. A concrete example: to solve −3x < 12, divide both sides by −3. Because you're dividing by a negative, flip the sign: x > −4.
You can always verify a solution by substituting a specific number. For x > −4, try x = 0: does −3(0) < 12? Yes, 0 < 12 ✓. Try x = −5: does −3(−5) < 12? That's 15 < 12, which is false ✗. So x = −5 correctly falls outside the solution set x > −4. This substitution check is your safeguard against sign-flip errors and direction-of-shading mistakes — two numbers, one inside and one outside the solution, catch both types of error at once.