# Inequalities that Describe Patterns

## Key Questions

• You need to follow the language properly.
Phrases such as 'at least' mean $\ge$ and phrases such as 'at most' or 'not more than' indicate $\le$.
'Less than' and 'more than' indicate $<$ and $>$ respectively.

Consider this problem-
You have gone to the market to buy gifts for more than 13 people. You have already bought 3 gifts. How many more do you need to buy?

See, the keyword here is 'more than'.
Let the remaining number of gifts be x. Therefore the required inequation will be 3+x>13, or, x>10.

See below:

#### Explanation:

An algebraic equality is when we have two statements and then say that they are equal. For instance:

$\frac{4}{2} = 2$ is an equality

$\frac{4}{2} = x$ is also an equality (and here we'd be looking for the value $x$)

An algebraic inequality is when there isn't a specific value or number where both sides equal each other. Instead, we'll be looking for a range of values that satisfy the statement. For instance:

$\frac{4}{2} < x$

We know that the value $x$ is all values less than 2 (there's an infinite number of solutions).

• Definitely

Most of the time, an inequality has more than one or even infinity solutions. For example the inequality: $x > 3$.

The solutions of this inequality are "all numbers strictly greater than 3". There's an infinite amount of these numbers (e.g. 4, 5, 100, 100000, 6541564564654645 ...). The inequality has an infinite amount of solutions. The notation for this is:
x in ]3;+oo[ or x in (3; +oo).
There will probably be more notations depending on in which country you live.