Linear Inequalities in One Variable
Solve and graph one-variable inequalities — and flip the sign on a negative.
The idea
A linear inequality is solved with the same moves as a linear equation, but its solution is a whole set of numbers: a ray, consisting of every number on one side of a boundary value $b$, with $b$ itself included when the relation is $\le$ or $\ge$ and excluded when it is strict.
To solve, isolate $x$. Add or subtract the same quantity on both sides, and multiply or divide both sides by the same nonzero number. One rule is new.
Proposition (Multiplying an inequality by a negative).
Let $a < b$. Adding the same number to both sides, or multiplying both sides by the same positive number, keeps the inequality: $a + c < b + c$, and $ac < bc$ when $c > 0$. Multiplying or dividing both sides by a negative number reverses it: $ac > bc$ when $c < 0$. The same holds with $\le$ and $\ge$ in place of
lt;$ and gt;$.
So dividing by a negative turns
lt;$ into gt;$ and $\le$ into $\ge$. The reason is that negation reverses order on the number line — $2 < 5$, but $-2 > -5$. Adding or subtracting shifts both sides the same distance and never disturbs the order.On a number line, draw the solution as a shaded ray, with a filled dot at an included boundary and a hollow dot at an excluded one.
Ways to work on it
- Walkthrough. Solve step by step and see when the sign flips.
- Practice. Solve a linear inequality and pick out its solution ray.
- Hardest. Variable on both sides, then graph the solution.
Not sure where to start? Take the ten-question placement test.