Inequalities

Inequalities: The Language of Comparison

When we say "Ravi has ₹200 and rice costs ₹30 per packet," we are not writing an equation. If Ravi buys $x$ packets, he spends $30x$ rupees. He cannot spend more than ₹200, but he also cannot spend exactly ₹200 unless 200 is a multiple of 30 — which it is not. So the relationship is $30x < 200$. This is an inequality: a statement that one quantity is less than (or greater than) another, not necessarily equal.

Similarly, if Reshma has ₹120, buys $x$ registers at ₹40 each and $y$ pens at ₹20 each, her total spending is $40x + 20y$. She can spend up to ₹120, so $40x + 20y \leq 120$. This single statement actually contains two possibilities: $40x + 20y < 120$ (she spends less than her money) and $40x + 20y = 120$ (she spends exactly all her money). The first is an inequality; the second is an equation.

Definition of an Inequality

Definition. Two real numbers or two algebraic expressions related by the symbol $<$, $>$, $\leq$, or $\geq$ form an inequality.

The four symbols have these meanings:

• $<$ : less than
• $>$ : greater than
• $\leq$ : less than or equal to
• $\geq$ : greater than or equal to

Numerical vs. Literal Inequalities

• Numerical inequalities involve only numbers: $3 < 5$, $7 > 5$.
• Literal inequalities involve variables: $x < 5$, $y > 2$, $x \geq 3$, $y \leq 4$.

Double Inequalities

A double inequality is a compact way to write two inequalities at once:

• $3 < 5 < 7$ means "5 is greater than 3 and less than 7."
• $3 \leq x < 5$ means "$x$ is greater than or equal to 3 and less than 5."
• $2 < y \leq 4$ means "$y$ is greater than 2 and less than or equal to 4."

Types of Inequalities: Strict vs. Slack

Inequalities are classified by whether the boundary value is included:

• Strict inequalities use $<$ or $>$ — the boundary value is not included.
• Slack inequalities use $\leq$ or $\geq$ — the boundary value is included.

Standard Forms of Linear Inequalities

The textbook lists the following standard forms. In each case, $a$, $b$, $c$ are real numbers, and $a \neq 0$ for one-variable forms, $a \neq 0$, $b \neq 0$ for two-variable forms.

Linear Inequalities in One Variable $x$

Linear Inequalities in Two Variables $x$ and $y$

Quadratic Inequalities (Not Linear)