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
- Solve before drawing Negative division reverses the sign. Verify the boundary and a value on each side in the original inequality
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
introRead strict rays with integer or half-integer boundaries
standardMatch inclusive inequalities to exact number-line rays
challengeSolve a negative-integer or fractional-coefficient inequality and choose its ray
examMatch or solve then match rays with all four inequality signs
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 a ray have a greatest integer but no greatest real value?+
Yes. For x < 3.5, the greatest integer is 3. There is no greatest real solution because an allowed value can always be chosen closer to 3.5
What does an arrow at the edge mean?+
The solution continues indefinitely in that direction. The edge of the drawing is not another boundary
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 8: Linear equations and inequalities
In this chapter
8A Linear equations and expressions: Solve a linear equationRecognise linear equations8A Linear equations and expressions: Set up and solve a practical linear equation8B Equations with brackets and fractions: Solve a linear equation8B Equations with brackets and fractions: Set up and solve a practical linear equationSolve equations that contain algebraic fractionsSubstitute values into a formulaSubstitute into and rearrange a formulaRead, draw and match inequality number lines8D Linear inequalities: Solve and interpret a linear inequationRead the point of intersection of two linesSolve a pair of simultaneous linear equationsSolve simultaneous equations by substitution8F Simultaneous equations by substitution: Solve a practical problem with simultaneous equations8F Simultaneous equations by substitution: Set up a pair of simultaneous equations from a description8F Simultaneous equations by substitution: Build and use a formula in two unknownsSolve simultaneous equations by elimination8G Simultaneous equations by elimination: Solve a practical problem with simultaneous equations8G Simultaneous equations by elimination: Set up a pair of simultaneous equations from a description8G Simultaneous equations by elimination: Build and use a formula in two unknowns8H Linear inequalities in the coordinate plane: Solve and interpret a linear inequation