How to read this expression? How can I read this expression :

$$\frac{1}{4} \le a \lt b \le 1$$ 

Means $a,b$ lies between $\displaystyle \frac{1}{4}$ and $1$?
Or is $a$ less the $b$ also less than equal to $1$?
So $a+b$ won't be greater than $1$?
 A: It means:
$$1/4\leq a\qquad \text{ and }\qquad a<b\qquad \text{ and }\qquad b\leq 1$$
You can combine this information to see that $a <1$ (because $a$ is strictly less than a number which is at most $1$) and $1/4<b$ (because $b$ is strictly greater than a number which is at least $1/4$).
A: Think of the number line.  The numbers $\{\tfrac{1}{4}, a, b, 1\}$ are arranged from left to right.  The weak inequalities on either end indicate that $a$ could be $\tfrac{1}{4}$ and $b$ could be $1$.  However, the strict inequality in the middle indicates that $a$ never equals $b$.
A: 
$$\frac{1}{4} \le a \lt b \le 1 \iff \;\; \frac 14 \leq a\;\;\;\text{AND}\;\;a\lt b\;\;\;\text{AND} \;\;b \leq 1$$ 

Most directly, this means that both $a$ and $b\;$ lie on the interval ranging from $\large \frac{1}{4}$ and $1$, inclusive, with the added qualification that $a\lt b$.
Another way to think of this inequality is to think of it as the set values, $\;a,\; b\;$ for which $$\;a \lt b\;\;\text{ and}\;\;\;[a, b]\; \subseteq \;[1/4, 1]$$
Yes, it could be that $a + b > 1$. For example, if $a = 1/2, \;b = 1$, then $\;a+b>1$
But it could also be that $a + b < 1:$  for example, if $a = 1/4, \; b = 1/2$, then the inequality is satisfied, but $a + b = 3/4 \lt 1$
