An equation can have one, zero or infinitely many solutions.
The equations with no solutions are:
We have:
[tex]2(x + 2) + 2 = 2(x + 3) + 1[/tex]
Open brackets
[tex]2x + 4 + 2 = 2x + 6 + 1[/tex]
[tex]2x + 6 = 2x + 7[/tex]
Subtract 2x from both sides
[tex]6 = 7[/tex]
The above equation is not true
i.e. [tex]6 \ne 7[/tex]
Hence:
[tex]2(x + 2) + 2 = 2(x + 3) + 1[/tex] has no solution
[tex]2x + 3(x + 5) = 5(x - 3)[/tex]
Open brackets
[tex]2x + 3x + 15 = 5x - 15[/tex]
[tex]5x + 15 = 5x - 15[/tex]
Subtract 5x from both sides
[tex]15 = - 15[/tex]
The above equation is not true
i.e. [tex]15 \ne - 15[/tex]
Hence:
[tex]2x + 3(x + 5) = 5(x - 3)[/tex] has no solution
[tex]4(x + 3) = x + 12[/tex]
Open bracket
[tex]4x + 12 = x + 12[/tex]
Subtract 12 from both sides
[tex]4x = x[/tex]
The above equation is not true
i.e. [tex]4x \ne x[/tex]
Hence:
[tex]4(x + 3) = x + 12[/tex] has no solution
[tex]4 - (2x + 5) = \frac 12(-4x - 2)[/tex]
Open brackets
[tex]4 - 2x + 5 = -2x - 1[/tex]
Add 2x to both sides
[tex]4 + 5 = - 1[/tex]
[tex]9 = -1[/tex]
The above equation is not true
i.e. [tex]9 \ne -1[/tex]
Hence:
[tex]4 - (2x + 5) = \frac 12(-4x - 2)[/tex] has no solution
[tex]5(x + 4) = 4(x + 5) -1[/tex]
Open brackets
[tex]5x + 20 = 4x + 20 -1[/tex]
[tex]5x + 20 = 4x + 19[/tex]
Collect like terms
[tex]5x - 4x = -20+ 19[/tex]
[tex]x=-1[/tex]
Hence,
Only [tex]5(x + 4) = 4(x + 5) -1[/tex] has a solution; others do not.
Read more about solutions of equations at:
https://brainly.com/question/18422820
