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
