Solving Inequalities on a Number Line

Definition An inequality is a statement that compares two real expressions with one of the order symbols $<$, $\le$, $>$ or $\ge$. For real numbers, $a < b$ means that $b - a$ is positive, and $a > b$ means the same as $b < a$. An inequality in a variable $x$ may be true for some real values of $x$ and false for others, so it makes sense to ask exactly which values make it true.

Definition The inequalities $a < b$ and $a > b$ are strict, and they exclude equality. The inequalities $a \le b$ and $a \ge b$ are weak (also called non-strict). The statement $a \le b$ means $a < b$ or $a = b$.

Definition The solution set of an inequality in $x$ is the set of all real numbers $x$ in the domain of both sides that make the inequality true. Two inequalities are equivalent when they have the same solution set. Solving an inequality means replacing it by simpler equivalent inequalities until the solution set can be read off.

Definition For all real numbers $a$, $b$ and $c$,

and if $c > 0$,

The same holds with $\le$ in place of $<$. Each of these steps can be undone by subtracting $c$ or by multiplying by $\frac{1}{c} > 0$, so they turn an inequality into an equivalent one.

Definition If $a < b$ and $c < 0$, then

So multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign, and $\le$ becomes $\ge$ in the same way. The reason is that $b - a > 0$ and $c < 0$ give $(b - a)c < 0$, which is $bc < ac$. Because of this, never multiply an inequality by an expression such as $x$ or $x - 1$ unless you know its sign. If the sign can change, split into cases where it is positive and where it is negative, and leave out the values where it is zero.

Definition A compound inequality such as $a < x < b$ is shorthand for two comparisons joined by “and”, here $a < x$ and $x < b$. A number is a solution exactly when it satisfies both, so the solution set is the intersection of the two separate solution sets. The same goes for mixed forms like $a \le x < b$. Such a chain is only written when $a < b$, since a chain like $5 < x < 2$ has no solutions at all.

Definition For real numbers $a < b$, interval notation names these sets.

A parenthesis marks an open endpoint that is left out, and a square bracket marks a closed endpoint that is included. The symbols $\infty$ and $-\infty$ are not real numbers, so they always get a parenthesis. The whole real line is $(-\infty, \infty)$, and a union of separate pieces is written with $\cup$.