Algebra - Linear Inequations

A Linear Inequation is a mathematical statement where two algebraic expressions are related by inequality symbols such as $<, >, \le,$ or $\ge$. Unlike an equation which has specific roots, an inequation represents a range of values that satisfy the condition.

The Replacement Set is the universal set from which the values of the variable $x$ are chosen (e.g., Natural Numbers $N$, Whole Numbers $W$, Integers $Z$, or Real Numbers $R$). The Solution Set is the specific subset of the replacement set containing all values that make the inequation true.

Rule of Addition and Subtraction: Adding or subtracting the same positive or negative number to both sides of an inequation does not change the direction of the inequality sign. For example, if $x < y$, then $x + a < y + a$ and $x - a < y - a$.

Rule of Multiplication and Division: Multiplying or dividing both sides by a positive number maintains the inequality. However, multiplying or dividing both sides by a negative number reverses the inequality sign (e.g., if $5 > 2$, then $-5 < -2$). Visually, this is because multiplying by a negative number reflects the points across the origin on the number line.

Representation on a Number Line for discrete sets ($N, W, Z$): Solutions are marked as distinct, bold dots on specific integer values. For instance, if $x \in \{1, 2, 3\}$, you would place bold dots only on the marks for $1, 2,$ and $3$ on the horizontal axis.

Representation on a Number Line for Real Numbers ($R$): Solutions are shown as a continuous thick line or shaded region. Use a hollow circle $\circ$ at the endpoint if the value is excluded ($<$ or $>$), and a solid darkened circle $\bullet$ if the value is included ($\le$ or $\ge$).

Combined Inequations (Double Inequalities): To solve a statement like $a \le f(x) < b$, split it into two separate inequations: $a \le f(x)$ and $f(x) < b$. Solve them independently and find the intersection of the two solution sets to determine the final range.