Question

How do you find the numbers such that four less than half the number is at least five and at most ten?

Answer 1

The number is any value between `18` and `28`, inclusive.

Explanation:

"`4` less than half the number" is written algebraically as :

`x/2 - 4`.

Since this expression "is at least `5`", it is greater than or equal to `5`, and since it is "at most `10`", it is less than or equal to `10`. Putting all this together gives the compound inequality:

`5 <= x/2 -4 <= 10`

Remember that to simplify this type of inequality, your objective is to isolate the variable in the center section of the statement. We will accomplish this by adding `4` to each section and then multiplying the entire inequality by `2`.

`5 + 4 <= x/2 - 4 + 4 <= 10 + 4`

`9 <= x/2 <= 14`

`2(9 <= x/2 <= 14)`

`18 <= x <= 28`

Therefore the number can be any number between `18` and `28`, inclusive.

Answer 2

`18<=x<=28` i.e `x` is between `18` and `28`

Explanation:

Let the number be `x`,

then four less than half the number is `x/2-4`

and as it is at least `5`, we have `x/2-4>=5`

or `x/2>=5+4` i.e. `x/2>=9` and `x>=18`

Further as `x/2-4` is at most `10`, we have `x/2-4<=10`

or `x/2<=10+4` i.e. `x/2<=14` and `x<=28`

Combining the two we have `18<=x<=28`

i.e `x` is between `18` and `28`