Inequality solutions and their representations
A one-variable inequality describes allowed values on a number line. A two-variable inequality describes allowed ordered pairs. Preserve the original comparison and apply any contextual domain restrictions
- Use inverse operations Add or subtract the same amount on both sides. Positive multiplication and division preserve the order
- Reverse for a negative factor Multiplication or division by a negative number reverses the inequality sign \(\displaystyle \begin{aligned}-7x&>21\\x&<-3\end{aligned}\)
- Transfer equation methods Clear a known positive denominator, undo or expand a bracket, and collect unknown terms on both sides. Apply each operation to both sides and retain the negative-factor reversal rule
- Include equality only when allowed A strict sign excludes the boundary. An inclusive sign includes it
- Check the original inequality Test the boundary and a value on the proposed solution side. A single successful test checks your result; it does not replace the algebra
- Distinguish integers from real numbers For an integer answer, select the first or last allowed integer. A strict real-number upper bound has no greatest real solution \(\displaystyle x<\frac{19}{2}\)
- Cancellation cases After the variable terms cancel, a true comparison means all real values and a false comparison means no real values
- Contextual domain Apply whole-number and non-negative constraints separately from the algebraic inequality. Verify the selected integer and the next excluded one
- Draw equality first Use two points for an oblique boundary. Treat x = k as vertical and y = k as horizontal
- Include or exclude equality Solid means included; dashed means excluded
- Test away from the line Substitute both coordinates into the original inequality. Select that side if true, the opposite side if false
- Respect context Add non-negative and whole-number restrictions when the variables represent counts. No optimisation algorithm is required here
Worked example
Represent 2x + y ≤ 4 and check which side is allowed. The diagram shows the boundary and test point before shading
- \(\displaystyle 2x+y=4\)Boundary points include (0,4) and (2,0). Use a solid line
- \(\displaystyle 2(0)+0\le4\)The origin is off the boundary and satisfies the original comparison
- \(\displaystyle y\le4-2x\)Shade the side below the line, containing the origin. The region is unbounded
Common mistakes
Difficulty levels
Questions about this topic
Does subtraction reverse the sign?+
No. Adding or subtracting the same number preserves the order. Negative multiplication or division reverses it
Is the boundary always a solution?+
Only an inclusive inequality allows equality at the boundary
Why might an integer answer differ from the boundary?+
The boundary may be excluded or may lie between integers. Choose an integer that belongs to the solution set
Should the sign reverse when I add a negative number?+
No. Adding or subtracting the same quantity preserves order. Multiplying or dividing by a negative number reverses it
What if all variable terms cancel?+
Evaluate the remaining comparison. If it is true, every real value is a solution; if false, no real value is a solution
Can I divide by the unknown?+
Only after considering its sign and excluding zero where division is used. A single unqualified division can lose cases or use the wrong comparison direction
Does a shaded region stop at the edge of the picture?+
No. Unless further constraints are stated, the region continues beyond the displayed window
Can I always use the origin as a test point?+
Only if it is not on the boundary. Otherwise choose another simple point away from the line
Does a vertical boundary use above or below?+
No. Use left or right: x compares horizontal coordinates while every y value may be allowed
Do I need linear programming?+
No. This lesson interprets constraints and checks feasible points without an optimisation algorithm
What do the different practice responses show?+
For a one-variable solution, enter its exact boundary and choose its relation; both must agree. Half-plane questions ask you to recognise an isolated inequality and a region description. Those choices do not assess an independently drawn graph. The squared-paper construction in the lesson is a self-check, not an automatically marked drawing
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 1: Linear relations and coordinate geometry