Question

How do you solve `abs(1 - 3b)= - 7`?

Answer 1

This equation has no solutions.

Explanation:

We have to start with the definition of the absolute value.
The absolute value of a non-negative number is this number itself.
Absolute value of the negative number is its negation.

In mathematical symbol it looks like that:
`X >= 0 => |X|=X`
`X < 0 => |X| = -X`

Using this definition, let's divide a set of all possible values of `b` into two parts:
(a) those where `1-3b >=0` (or `b<=1/3`)
(b) those where `1-3b < 0` (or `b>1/3`).

In case (a) our equation looks like this:
`1-3b = -7`,
which has a solution `b=8/3`.
This solution does not belong to the area of `b<=1/3` and must be discarded.

In case (b) our equation looks like this:
`-(1-3b) = -7`,
which has a solution `b=-2`.
This solution does not belong to the area of `b>1/3` and must be discarded.

So, no solutions are found for this equation.
We can confirm this graphically by observing that function `y=|1-3x|+7` does not have intersections with X-axis.

graph{|1-3x|+7 [-46.23, 46.25, -23.12, 23.1]}

Answer 2

exactly no solutions

Explanation:

because
`absa` can't be negative,
you can't find any solution which satifies this equation