To solve the absolute value inequality |4x+1| < 21, we need to isolate the absolute value expression and split the inequality into two separate cases.
1. When 4x + 1 is positive:
In this case, the absolute value is equal to the expression itself. So, we can write:
4x + 1 < 21
To solve this inequality, we can subtract 1 from both sides:
4x < 20
Next, divide both sides by 4:
x < 5
2. When 4x + 1 is negative:
In this case, the absolute value is the negation of the expression. So, we can write:
-(4x + 1) < 21
To solve this inequality, we first distribute the negative sign:
-4x - 1 < 21
Next, add 1 to both sides:
-4x < 22
Then, divide both sides by -4. Since we are dividing by a negative number, we need to reverse the inequality sign:
x > -5.5
So, the solution to the absolute value inequality |4x+1| < 21 is:
x < 5 or x > -5.5
This means that x can be any value less than 5 or any value greater than -5.5. For example, x = 4 or x = -4 would satisfy the inequality.