7.1 Inequalities
An inequality is a mathematical statement comparing two quantities with different values using inequality symbols.
Inequality Symbols
- > : Greater than
- < : Less than
- ≥ : Greater than or equal to (At least / Minimum)
- ≤ : Less than or equal to (At most / Maximum)
Properties of Inequalities
- Converse Property: If $a > b$, then $b < a$. If $a < b$, then $b > a$.
- Transitive Property: If $a < b$ and $b < c$, then $a < c$.
- Addition & Subtraction: If $a < b$, then $a + c < b + c$ and $a - c < b - c$. (Inequality sign remains unchanged)
- Multiplication & Division by a Positive Number: If $a < b$ and $c > 0$, then $a \times c < b \times c$ and $\frac{a}{c} < \frac{b}{c}$. (Inequality sign remains unchanged)
- Multiplication & Division by a Negative Number: If $a < b$ and $c < 0$, then $a \times c > b \times c$ and $\frac{a}{c} > \frac{b}{c}$. (The inequality symbol MUST be reversed!)
7.2 Linear Inequalities in One Variable
A linear inequality in one variable is an inequality involving only one variable whose highest power is 1.
General Form Examples:
- $x + 3 > 7$
- $2x - 5 \le 9$
- $\frac{y}{4} + 1 \ge -2$
Solving Linear Inequalities
Solving a linear inequality means finding all values of the variable that make the statement true. Follow basic algebraic operations, but remember to reverse the inequality symbol whenever you multiply or divide both sides by a negative number.
Number Line Representation
- Empty Circle ($\circ$): Used for strictly greater than ($>$) or strictly less than ($<$). The endpoint is excluded.
- Solid Circle ($\bullet$): Used for greater than or equal to ($\ge$) or less than or equal to ($\le$). The endpoint is included.
- Arrow Direction: Points right for greater values ($>, \ge$) and left for smaller values ($<, \le$).
7.3 Simultaneous Linear Inequalities in One Variable
When two or more linear inequalities in the same variable are solved together, the solution set consists of the values that satisfy all the inequalities simultaneously (the common region on the number line).
Steps to Solve Simultaneous Inequalities:
- Solve each linear inequality independently.
- Draw both solutions on a single shared number line.
- Identify the overlapping region (intersection) satisfied by both inequalities.
- Write down the combined linear inequality range.