Answer:
To determine the number of solutions, we have to solve each equation for x.
1.- To solve the equation for x, first, we apply the distribute property on both sides of the equation:
[tex]2x-14=-3x+18.[/tex]Adding, 3x to both sides of the equation, we get:
[tex]\begin{gathered} 2x-14+3x=-3x+18+3x, \\ 5x-14=18. \end{gathered}[/tex]Adding 14 we get:
[tex]\begin{gathered} 5x-14+14=18+14, \\ 5x=32. \end{gathered}[/tex]Dividing by 5, we get:
[tex]x=\frac{32}{5}\text{.}[/tex]Therefore, the equation has one solution.
2.- Applying the distributive property on the right side of the equation, we get:
[tex]14-3x-5=3x+9.[/tex]Adding like terms, we get:
[tex]9-3x=3x+9,[/tex]therefore, by properties of real numbers,
[tex]x=0.[/tex]The equation has 1 solution.
3.- Applying the distributive property on the right side of the equation, we get:
[tex]12x-1=12x-6.[/tex]Subtracting 12x from both sides of the equation, we get:
[tex]\begin{gathered} 12x-1-12x=12x-6-12x, \\ -1=-6. \end{gathered}[/tex]The above equality is a contradiction, therefore the equation has no solutions.
4.- Applying the distributive property on the left side of the equation, we get:
[tex]x+5x=6x\text{.}[/tex]Simplifying the above equation, we get:
[tex]6x=6x\text{.}[/tex]Since the above equality is always true for any value of x, then the equation has infinitely many solutions.
5.- Applying the distributive property on the left side of the equation, we get:
[tex]36-8x-2x=8-10x\text{.}[/tex]Adding like terms, we get:
[tex]36-10x=8-10x\text{.}[/tex]Adding 10x to both sides of the equation, we get:
[tex]\begin{gathered} 36-10x+10x=8-10x+10x, \\ 36=8. \end{gathered}[/tex]The above equality is a contradiction, therefore the equation has no solutions.