Key Points

An inequality is a statement that two real numbers or two algebraic expressions are related by one of the symbols $<$, $>$, `$\leq$`, or `$\geq$`. For example,$3x+5 < 10$ is a linear inequality.

Inequalities using the symbols $<$ (less than) or $>$ (greater than) are called strict inequalities. Inequalities using the symbols $\leq$ (less than or equal to) or $\geq$ (greater than or equal to) are called slack inequalities.

An inequality that can be written in the form $ax+b < 0$, $ax+b > 0$, $ax+b \leq 0$, or $ax+b \geq 0$, where $a$ and $b$ are real numbers and $a \neq 0$, is a linear inequality in one variable.

An inequality that can be written in the form $ax+by < c$, $ax+by > c$, $ax+by \leq c$, or $ax+by \geq c$, where $a, b, c$ are real numbers and $a \neq 0, b \neq 0$, is a linear inequality in two variables.

Any value of the variable which makes the inequality a true statement is called a solution. The set of all solutions is called the solution set.

Equal numbers can be added to or subtracted from both sides of an inequality without changing the sign of inequality. If $a < b$, then $a+c < b+c$.

Both sides of an inequality can be multiplied or divided by the same positive number without changing the sign of inequality. If $a < b$ and $c > 0$, then $ac < bc$.

When both sides of an inequality are multiplied or divided by a negative number, the sign of inequality is reversed. If $a < b$ and $c < 0$, then $ac > bc$.

To solve a double inequality like $a < bx+c < d$, apply the same operations to all three parts to isolate the variable $x$ in the middle. For example, $-8 \leq 5x-3 < 7$ becomes $-5 \leq 5x < 10$, which simplifies to $-1 \leq x < 2$.

Solutions to one-variable inequalities are shown on a number line. A hollow circle is used for strict inequalities ($<$ or $>$), and a solid circle is used for slack inequalities ($\leq$ or $\geq$).

To solve a system of inequalities, find the solution set for each inequality separately. The final solution is the intersection of these individual solution sets, representing the values that satisfy all inequalities simultaneously.

The phrase 'at least' translates to the $\geq$ symbol, while 'at most' translates to the $\leq$ symbol. For example, an average of at least 60 means the average must be $\geq 60$.