Question

Absolute value inequality of |4x+1| < 21

Answer 1

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.

Answer 2

`|4x+1| < 21\\4x+1 < 21 \wedge 4x+1 > -21\\4x < 20 \wedge 4x > -22\\x < 5 \wedge x > -5.5\\x\in(-5.5,5)`