Solving inequalities

An inequality compares two sides using $<$, $>$, $\leq$, or $\geq$ instead of $=$.
To solve it, you find all the values of $x$ that make it true.
The steps are the same as solving an equation: gather the $x$-terms on one side and the numbers on the other.

Take $3x + 5 < 2x + 10$.
Subtract $2x$ from both sides: $x + 5 < 10$.
Subtract $5$ from both sides: $x < 5$.
So every number less than $5$ is a solution.

Here is the one rule that is different from equations.
If you multiply or divide both sides by a negative number, flip the inequality sign.
For example, $-x \geq 4$.
Multiply both sides by $-1$, and the $\geq$ becomes $\leq$, giving $x \leq -4$.

The answer to an inequality is usually a range, not a single number.
On a number line, an open circle means the endpoint is not included (for $<$ or $>$).
A filled circle means the endpoint is included (for $\leq$ or $\geq$).

A system of inequalities is two or more inequalities at once.
A point is a solution only if it satisfies every inequality in the system.
If it fails even one, it is not a solution.

Solve $2x + 3 < x + 9$.
Subtract $x$ from both sides: $x + 3 < 9$.
Subtract $3$ from both sides: $x < 6$.

Solve $4 - 2x \geq 10$.
Subtract $4$ from both sides: $-2x \geq 6$.
Divide both sides by $-2$ and flip the sign: $x \leq -3$.

Solve $-3x < 12$.
Divide both sides by $-3$ and flip the sign: $x > -4$.
So every number greater than $-4$ is a solution.