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?

1 Answer
May 7, 2018

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#