Inequalities on number lines
Read the boundary values, the endpoint styles and the shaded region. A ray continues indefinitely. In these finite lesson drawings, the question states when the shading continues beyond the displayed window
- Strict signs A hollow endpoint excludes equality \(\displaystyle x<9\quad\text{or}\quad x>9\)
- Inclusive signs A filled endpoint includes equality \(\displaystyle x\le9\quad\text{or}\quad x\ge9\)
- Ray direction Values smaller than the boundary are left of it. Values greater than the boundary are right of it
- Bounded intervals Shade between the two boundaries and decide each endpoint style separately \(\displaystyle -7\le x<1\)
- Integers and real values An interval can include non-integer real values. Count only integer members when the question asks for integers
Worked example
Describe the number line that represents this inequality
\[z<-6\]
- \(\displaystyle z=-6\)Place the boundary at negative six
- \(\displaystyle -6\not<-6\)Equality does not satisfy the strict sign. Use a hollow endpoint
- \(\displaystyle -7<-6\)A smaller value satisfies the inequality. Shade a ray to the left
- \(\displaystyle z<-6\)The ray contains all values below the boundary and excludes the boundary itself
Common mistakes
Excluding a filled endpoint
✕\(\displaystyle x\ge5\Rightarrow\text{hollow endpoint}\)✓\(\displaystyle x\ge5\Rightarrow\text{filled endpoint}\)
An inclusive sign allows equalityShading the opposite side
✕\(\displaystyle x<-7\Rightarrow\text{shade right}\)✓\(\displaystyle x<-7\Rightarrow\text{shade left}\)
Smaller values lie to the left, including on the negative side of zeroIgnoring the upper boundary
✕\(\displaystyle -8<x\le-1\Rightarrow x>-8\)✓\(\displaystyle -8<x\le-1\)
Both restrictions apply. Use a bounded segment rather than an unrestricted rayDifficulty levels
introMatch strict inequalities and number-line rays in both directions, with small signed integer boundaries
standardMatch inequalities and rays in both directions, mixing all four signs with positive, zero and negative boundaries
challengeRead and match bounded intervals in both directions, with each endpoint independently included or excluded
examChoose the matching inequality or diagram for a ray or bounded interval
Questions about this topic
What does a hollow endpoint mean?+
The boundary itself is excluded. Values can still lie arbitrarily close to it on the shaded side
Does a negative boundary change the shading rule?+
No. Smaller values are always to the left and larger values are always to the right
Can a bounded interval have different endpoint styles?+
Yes. Check each boundary separately; one can be included while the other is excluded
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 2: Linear equations
In this chapter
Translate between a description and an algebraic expressionEvaluate an algebraic expression by substitutionTranslate between a description and an algebraic expressionSimplify algebraic terms and collect like termsSimplify algebraic terms and collect like termsExpand brackets and collect like terms2D Two-step linear equations: Solve a linear equation2E Linear equations with brackets: Solve a linear equationSet up and solve a practical linear equationSolve and interpret a linear inequationRepresent and read inequalities on a number lineSubstitute values into a formulaSolve for an unknown or recognise a rearranged formulaSolve a pair of simultaneous linear equations (optional extension)Solve a pair of simultaneous linear equations by elimination (optional extension)Set up a pair of simultaneous equations from a description (optional extension)Solve a practical problem with simultaneous equations (optional extension)