Question

how to draw the graph for the absolute value inequality : |4x+1| < 5

Answer 1

i am not a pro at this, but i did do some similar problems like this.
absolute value is just like the number of units you're away from 0.
you don't have to think about the two bar signs there.
just compute it as 4x + 1 = 5
then carry down the value of x, the sign, and (i believe 5).
finally show it in your graph :)

Answer 2

Remember

|x + a| < c is equivalent to -c < x + a < c which is also equivalent to
-c - a < x < c - a

Answer 3

In other words, if you had for example |2x + 4| < 7, you would perform the following steps to isolate x:

First convert |2x + 5| < 7 to it's first equivalent form:
-7 < 2x + 5 < 7

Then convert that form to the next equivalent form:
-7 - 5 < 2x < 7 - 5
-12 < 2x < 2

Then finally isolate x by dividing each side by 2:
-12/2 < x < 2/2
-6 < x < 1

Answer 4

From there, it is easy to see that x is everything value between -6 and 1. And now we can simply make a number line to show this: