Multi-step inequalities are solved using the same techniques as multi-step equations — distribute, combine like terms, use inverse operations — with the added rule that multiplying or dividing both sides by a negative number reverses the inequality sign. The solution is a range of values, graphed on a number line with open or closed circles and shading. For example, −3x + 7 > 1 becomes −3x > −6, then x < 2 (sign flipped because of division by −3). Inequalities model real-world constraints: budgets, speed limits, minimum requirements.
Solve the corresponding equation first to find the boundary value, then determine the direction of the inequality by testing a point. This reinforces the equation-inequality connection. Practice the sign-flip rule extensively with dedicated exercises. Graph every solution on a number line and verify by substituting a value from the solution region into the original inequality.
Solving a multi-step inequality is almost identical to solving a multi-step equation — you isolate the variable using the same inverse-operation strategy you learned with equations. The key insight is that an inequality doesn't give you a single answer; it describes an entire region of values that satisfy a condition. Think of it as asking: "For which values of x is this statement true?" The answer is always an interval or ray on the number line, not just one point.
The mechanics mirror equation-solving closely. Take −3x + 7 > 1. Subtract 7 from both sides: −3x > −6. Now divide both sides by −3 — and here is the one new rule. Dividing or multiplying both sides of an inequality by a negative number reverses the direction of the inequality sign. This happens because multiplying by −1 flips the number line: what was larger becomes smaller. So −3x > −6 becomes x < 2. The inequality flipped from > to <.
Why does the flip happen? Imagine the true inequality 3 > 1. Multiply both sides by −1 and you get −3 and −1. On the number line, −3 is to the *left* of −1, meaning −3 < −1. The relationship reversed. This same logic applies whenever a negative factor appears. A reliable strategy: solve the corresponding equation first to find the boundary value (x = 2 here), then test one point on each side to determine which region satisfies the original inequality.
The solution x < 2 is graphed on a number line with an open circle at 2 (the boundary is not included because the inequality is strict) and shading extending to the left. Had the inequality been ≤ instead of <, the circle would be closed. This graphical representation communicates the full solution set at a glance. Verify your answer: pick x = 0 (inside the solution region): −3(0) + 7 = 7 > 1 ✓. Pick x = 3 (outside): −3(3) + 7 = −2, and −2 > 1 is false ✓. Both checks confirm the solution.