Solving $x<\frac y2$ and $x+y>60$ Hi the question is as follows:
John is less than half his mother's age, though the sum of their ages is greater than 60. John was born on his mother's 26th birthday.
By expressing these facts as two inequalities and one equation, find the range of possible ages for John.
When trying I always seem to get a contradiction but then also miss out what the top of the range is.
Thanks in advance
 A: You missed a crucial piece of information, which was "John was born on his mother's 26th birthday." If we let John's age be equal to $x$, and John's mother's age be equal to $y$, this becomes $x = y - 26$. We can use this equation, as well as the other two inequalities, to solve this problem.
$x = y - 26$ and $x + y > 60$ together imply $x + (x + 26) > 60$, which simplifies to $x > 17$.
$x = y - 26$ and $x < y/2$ together imply $x < (x + 26)/2$, which simplifies to $x < 26$. 
Hence, the range of possible ages for John is $17 < x < 26$.
A: Observe that we are given $$x<\frac y2\qquad x+y>60\qquad x+26=y$$ Thus, substituting $y$ in the inequalities yields $$2x<x+26\iff x<26\qquad 2x+26>60\iff x>17$$
Therefore 

$$x\in\{n\in\Bbb N: 18\le n\le 25\}$$

A: Since John was born on his mother's 26th birthday you know that
$y-x = 26 $ or $ y=x+26$
You are given that
$x<y/2  \implies  x<(x+26)/2 \implies 2x <x+26 \implies x<26  .....(1)$
You are also given that
$x+y>60 \implies x+x+26>60 \implies 2x>34 \implies x>20  .....(2)$
From these 2 ranges you know that John's age is between 17 and 26.
