How would I solve the inequality 7<4x+3<21 and 
express the solution as an interval?

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 7 < 4x + 3 < 21

The idea is to isolate x in the middle.
Get rid of the 3 by adding -3 to all three sides:

  7 < 4x + 3 < 21
 -3       -3   -3
-------------------
  4 < 4x     < 18

So we have
 
    4 < 4x < 18

Now we divide all three sides by the coefficient of x,
which is 4.  Since we are dividing by a positive
number we DO NOT reverse the inequalities as we do
when we divide thru by a negative number:

    4     4x     18
   ——— < ———— < ————
    4      4      4

or
                 9  
        1 < x < ———
                 2
or

        1 < x < 4.5



On a number line this is graphed as

--------------o=============·------------
 -2  -1   0   1   2   3   4   5   6   7          

It is an open circle at 1, because
it is < and not <, and closed (dark
circle) at 4.5, because it is < and not <.

The interval notation is a shorthand
for this graph:

       (1, 4.5]

The "(" on the left represents the
open circle, the 1 is the left endpoint
of the shaded part, the 4.5 is the right endpoint
of the shaded part, and the "]" represents
the closed circle.

Edwin