Question

Half the sum of a number, x, and 15 is at most the sum of the opposite of twice the number and 1.25. What is the range of possible values for the number?

Answer 1

Any value smaller than `-2.5` (included) is fine: the range is

`R = {x \in \mathbb{R}: x \leq -2.5}`

Explanation:

If you know that `A` is at most `B`, it means that `A` can't be more than `B`, and thus is lesser or equal to `B`. So, we will have something like `A \leq B`.

Let's translate `A` and `B`:

A:

• The sum of a number `x` and `15 \implies x+15`
• Half the sum of a number `x` and `15 \implies \frac{x+15}{2}`

B:

• The opposite of the number `implies -x`
• The opposite of twice the number `\implies -2x`
• The sum of the opposite of twice the number and `1.25 \implies -2x+1.25`

So, we have

`\frac{x+15}{2} \leq -2x+1.25`

Multiply both sides by two:

`x+15 \leq -4x+2.5`

Bring all the `x` terms of the left and the numbers to the right:

`5x \leq -12.5`

Solve for `x`:

`x \leq -\frac{12.5}{5} = -2.5`