What Inequalities Are | Moosa Academy

An equation pins a variable to a single value. An inequality does something different: it describes a whole range of values at once — and on the coordinate plane, a whole region rather than a line.

Concept Equation against inequality

x = 5

has exactly one solution: the number 5.

x > 3

has infinitely many: 3.1, 4, 100, and everything beyond.

That is the whole distinction. An equals sign identifies a point; an inequality sign opens up an interval.

Concept The four symbols

>

greater than, as in

x > 3

<

less than, as in

x < 5

\geq

greater than or equal to, as in

x \geq 3

\leq

less than or equal to, as in

x \leq 5

The bar underneath is what decides whether the boundary value itself counts as a solution.

Example On the number line

For

x > 3

the boundary is excluded, marked by a hollow circle. For

x \geq 3

it is included, marked by a filled one. The shaded ray is identical in both cases.

Concept Inequalities in two dimensions

With two variables, the equation

y = 2x + 3

draws a line. The inequality

y > 2x + 3

shades everything above it, while

y < 2x + 3

shades everything below.

The same rule about the boundary applies: a strict sign gives a dashed line, and $\geq$ or $\leq$ gives a solid one.

Example Four regions from one line

Take the boundary $y = 2x + 3$ . Four inequalities can be built from it:

y > 2x + 3

— above, dashed line.

y \geq 2x + 3

— above, solid line.

y < 2x + 3

— below, dashed line.

y \leq 2x + 3

— below, solid line.

⟹ direction sets the side, the bar sets the line style

Summary

An equation gives one solution; an inequality gives a range.
The symbols are >, <, ≥ and ≤.
With > or < the boundary is excluded: hollow circle, dashed line.
With ≥ or ≤ the boundary is included: filled circle, solid line.
In two variables an inequality shades a region, not a line.