One-Variable Inequalities
Solve a one-variable inequality by preserving order, then interpret the result as an entire region of allowed values instead of one exact answer.
Why does an inequality usually have a whole region of answers?
An equation asks where two quantities are equal. An inequality asks where one stays ordered relative to another. Once the variable is isolated, a boundary divides the number line into allowed and disallowed regions.
The equality bar decides whether the boundary value itself belongs to the region.
Solve first. Interpret the region second.
For 2x + 1 < 7, the algebra follows the same balance logic as an equation. The difference appears at the end: x < 3 names every value to the left of 3.
Subtract 1 from both sides.
Divide both sides by positive 2. The order stays the same.
3 is the boundary; all smaller values satisfy the inequality.
Multiplying by a negative reflects the number line.
Start with 2 < 5. Multiply both values by −1 and they become −2 and −5. Reflection across zero swaps left and right, so the equivalent comparison is −2 > −5. The symbol reversal records that geometric change in order.
Preserve the order while you isolate x.
Remove the +1 from both sides.