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

Jun 17, 2016

This equation has no solutions.

#### Explanation:

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:
$X \ge 0 \implies | X | = X$
$X < 0 \implies | X | = - X$

Using this definition, let's divide a set of all possible values of $b$ into two parts:
(a) those where $1 - 3 b \ge 0$ (or $b \le \frac{1}{3}$)
(b) those where $1 - 3 b < 0$ (or $b > \frac{1}{3}$).

In case (a) our equation looks like this:
$1 - 3 b = - 7$,
which has a solution $b = \frac{8}{3}$.
This solution does not belong to the area of $b \le \frac{1}{3}$ and must be discarded.

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

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

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

Jun 17, 2016

exactly no solutions

#### Explanation:

because
$\left\mid a \right\mid$ can't be negative,
you can't find any solution which satifies this equation