Inequalities using Set Notation
# Inequalities using Set Notation

GCSE(H),

Inequalities can be shown using set notation. The set notation is shown in the form:

{`x`: *inequality* }

where `x:` indicates the variable being described and *inequality* is written as an inequality, normally in its simplest form. The colon means *such that*.

To describe `4x + 5 > 25` in set notation, simplify to `x > 5` and show as `{x: > 5}`. A description of the set notation is *a set of numbers such that `x > 5`*.

A more complex inequality would be shown as `{x: -5 < x < 5}`.

The inequalities can be shown in combinations, using set notation such as the union of two sets:

`{x: < 4} uu {x: > 6}`

## Examples

1. Write `x^2 - 4 > 0` in its simplest form in set notation.

Answer: `{x: > 2} nn {x: < -2}`

Solve: `x^2 - 4 > 0` to get `x^2 > 4`. Square rooting gives two solutions:
`x` must be greater than 2; or `x` must be less than -2:
`{x: > 2} nn {x: < -2}`.

2. Which integers are described by this set description?

`{x: > 4} nn {x: -3 ≤ x < 9}`

Answer: 5, 6, 7, 8 and 9

The first set describes all integers that are 5 or greater. The seconds set describes integers from -3 to 9 inclusive. Integers in both sets are 5, 6, 7, 8 and 9.

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