Question

How do you solve `1/4|2x+5|+1=7`?

Answer 1

There are two solutions:
`x_1=23/2`
`x_2=-33/2`

Explanation:

Start from definition:
`a >= 0 => |a| = a`
`a < 0 => |a| = -a`

Using this, divide all the real numbers into two groups:
`2x+5 >= 0`, that is `x >= -5/2` and
`2x+5 < 0`, that is `x < -5/2`

Case 1. Looking for solutions among `x >= -5/2`
Our equation can be written as
`1/4(2x+5) = 7` or
`2x+5 = 28`
Solution is `x = 23/2`
This value is greater than `-5/2` and, therefore, is a legitimate solution.

Case 2. Looking for solutions among `x < -5/2`
Our equation can be written as
`-1/4(2x+5) = 7` or
`2x+5 = -28`
Solution is `x = -33/2`
This value is less than `-5/2` and, therefore, is a legitimate solution.

CHECK
1. `1/4|2*23/2+5|=1/4*|28|=28/4=7`
2. `1/4|2(-33/2)+5|=1/4|-28|=28/4=7`

You can see these two solution on a graph of
`y=1/4|2x+5|-7`
that intersects an X-axis (that is, equals to zero) at the solutions of the initial equation.
graph{1/4|2x+5|-7 [-18.02, 18.02, -9.01, 9.01]}