FOUNDATIONS 04
Inequalities
Inequalities describe order and allowed regions: intervals on a number line, areas in a plane, and regions of space.
The main idea
An equation such as \(x=4\) identifies an exact boundary or value. An inequality such as \(x\le4\) describes all values on one side of that boundary, including or excluding the boundary depending on the symbol.
\[\boxed{\text{equation: boundary or exact value}\qquad\text{inequality: allowed region}}\]The same idea extends naturally from one dimension to two and three dimensions.
1. The inequality symbols
The four basic symbols compare two quantities.
a is less than b
a is greater than b
a is at most b
a is at least b
2. Strict inequalities: boundary excluded
\(x\gt2\) means all real numbers greater than \(2\), but not \(2\) itself.
An open circle has no colour inside: the endpoint is not included.
3. Inclusive inequalities: boundary included
\(x\le4\) includes every value less than \(4\) and also includes \(4\) itself.
A filled circle shows that equality is allowed.
4. Adding or subtracting preserves order
Adding the same quantity to both sides shifts both values equally, so their order remains unchanged.
\[a\lt b\Longrightarrow a+c\lt b+c.\]\(x+3-3\lt8-3\)
\(\boxed{x\lt5}\)
5. Multiplying by a positive number preserves order
If \(c\gt0\),
\[a\lt b\Longrightarrow ac\lt bc.\]Positive scaling does not reverse the number line.
multiply by \(3\)
\(6\lt15\)
6. Multiplying by a negative number reverses order
Multiplication by \(-1\) reflects the number line about zero. Thus \(2\lt5\) becomes \(-2\gt-5\).
Therefore multiplying or dividing an inequality by a negative number reverses the sign.
divide by \(-2\)
\(\boxed{x\gt-3}\)
7. Double inequalities
\(2\lt x\lt5\) means that both inequalities are true:
\[x\gt2\qquad\text{and}\qquad x\lt5.\]Both endpoints are excluded, so both circles are open.
8. Mixed endpoints and interval notation
For \(2\le x\lt5\), the left endpoint is included and the right endpoint is excluded.
\[2\le x\lt5\quad\Longleftrightarrow\quad x\in[2,5).\]Inequalities in two dimensions
With two variables, an equation usually forms a curve or line, while an inequality selects a region on one side of that boundary.
9. A linear inequality in the plane
Consider
\[y\le x+1.\]The boundary is the line \(y=x+1\). Because equality is allowed, the boundary line is solid. The solution is every point on or below that line.
10. Strict inequality in the plane
For
\[y\gt x+1,\]the same line is the boundary, but equality is not allowed. Therefore the boundary is drawn dashed and the solution lies strictly above it.
11. Testing which side of a boundary is correct
A boundary divides the plane into regions. Choose a simple test point not on the boundary, often \((0,0)\), and substitute it.
For \(y\le x+1\),
\[0\le0+1\quad\text{is true}.\]So the side containing \((0,0)\) is part of the solution.
\(0\le0+1\)
\(0\le1\quad\checkmark\)
shade the side containing \((0,0)\)
12. A system of inequalities gives an intersection
Consider
\[x\ge0,\qquad y\ge0,\qquad x+y\le4.\]Each inequality gives a half-plane. The points satisfying all three form the triangular feasible region.
Inequalities in three dimensions
The same principle continues in space. An equation in three variables can form a surface, and an inequality chooses one side of that surface.
13. A plane divides three-dimensional space
The equation
\[x+y+z=4\]defines a plane. The inequality
\[x+y+z\le4\]selects the half-space on one side of that plane, including the plane itself.
This becomes important later in multivariable calculus and constrained optimisation.
In one dimension, inequalities describe intervals. In two dimensions, they describe regions bounded by lines or curves. In three dimensions, they describe regions bounded by surfaces. A strict inequality excludes the boundary; an inclusive inequality includes it. In diagrams, excluded endpoints are hollow and included endpoints are filled, while strict 2D boundaries are dashed and inclusive boundaries are solid.
Why this matters for calculus
Inequalities define domains, intervals of increase and decrease, error bounds in limits, feasible regions in optimisation, and regions of integration. The move from intervals to areas and volumes is exactly the move from single-variable calculus to multivariable calculus.