Solving a One-Variable Inequality

An inequality compares two expressions with greater than, less than, or their 'or equal to' versions. You solve it the same way you solve an equation: add, subtract, multiply, divide to get x alone. The one difference that matters is what happens when you multiply or divide both sides by a negative number. Numbers keep their order under addition and subtraction, but multiplying by a negative reflects every number across zero, so the larger one lands on the left and the smaller one lands on the right. The symbol has to turn around to keep describing the truth.

Picture a number line with 2 and 5 on it, 2 to the left since it's smaller. Multiply both by -1 and they land at -2 and -5. Now -5 is to the left, meaning -5 is smaller, so the relationship reversed even though you did the same thing to both sides. That physical flip is why the rule exists: it isn't arbitrary, it's just bookkeeping for a reflection.

This matters most in word problems and systems built from inequalities, where a careless sign flip gives a solution set that's the mirror image of the right one. It also connects to compound and absolute value inequalities, where you apply the same flip rule to each piece, and to graphing, where the boundary value becomes an open or closed circle or a dashed or solid line depending on whether equality is allowed.