2
Unbounded intervals with infinity: positive, negative, at least, at most
Mathematics Prep 2 Term 1, Unit 1, Lesson 2
GoalWrite unbounded intervals with ∞ and −∞, and know R, R⁺ and R⁻.
New words
infinity (∞) · a symbol for ‘goes on for ever’; it is not a numberunbounded interval · an interval with no end on one side or on both sidesR⁺ and R⁻ · the positive reals ]0, ∞[ and the negative reals ]−∞, 0[at least / at most · x ≥ a and x ≤ a
Explanation
Some intervals go on for ever. x > 2 is the interval ]2, ∞[ and x ≥ 2 is [2, ∞[. In the same way x < 1 is ]−∞, 1[ and x ≤ 1 is ]−∞, 1]. The whole number line is R = ]−∞, ∞[, the positive real numbers are R⁺ = ]0, ∞[ and the negative ones are R⁻ = ]−∞, 0[. The symbol ∞ is not a number, so you can never reach it: its bracket is always open, and ∞ ∉ R. ‘At least 12’ means 12 or more, so it is [12, ∞[; ‘at most 5’ means 5 or less, so it is ]−∞, 5]. Note that 0 belongs to neither R⁺ nor R⁻.
| Inequality | Interval |
|---|---|
| x > a | ]a, ∞[ |
| x ≥ a | [a, ∞[ |
| x < a | ]−∞, a[ |
| x ≤ a | ]−∞, a] |
Worked examples
Show [2, ∞[ on the number line. The filled dot is at 2 because 2 is included, and the arrow goes right for ever. In set form: {x : x ∈ R, x ≥ 2}.
Write x < 1 as an interval and draw it. It is ]−∞, 1[. The empty circle is at 1 and the arrow goes left. The number 1 is not in the set, but 0.999 is.
Write each situation as an interval. A ride is for children 120 cm tall or more: [120, ∞[. A freezer must be at −18 °C or colder: ]−∞, −18]. A fridge is set between 2 °C and 5 °C, both included: [2, 5].
Class activity
Real-life intervals. Collect five rules from daily life, such as ‘children under 6 travel free’ or ‘the lift carries at most 8 people’. Write each as an interval. Is the end point included? Is the interval bounded?
Practice
1
Write x > −3 as an interval.
]−3, ∞[.
2
Write [4, ∞[ as an inequality and in set form.
x ≥ 4, or {x : x ∈ R, x ≥ 4}.
3
Is 0 ∈ R⁺ ? Is ∞ ∈ R ?
No to both. R⁺ = ]0, ∞[ leaves out 0, and ∞ is not a number.
4
A lift carries at most 8 people. Write the number of people as an interval of real numbers, starting from 0.
[0, 8].
5
Write the interval shown.
]−∞, 2]: the filled dot at 2 and the arrow to the left.