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

2 Answers
Jun 17, 2016

Answer:

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]}

Jun 17, 2016

Answer:

exactly no solutions

Explanation:

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