How many solutions exist for the given equation? 0.75(x+40)=3(4x+1)-2

The given equation is
0.75(x + 40) = 3(4x + 1) - 2

Expand the equation.
0.75x + 30 = 12x + 3 - 2
0.75 - 12x = 3 - 2 - 30
-11.25x = -29
x = -29/-11.25 = 2.5778

There is 1 solution, which is 2.5778.

Let's eliminate the fractional multiplier 0.75.  To do this, mult the entire equation by 4/3:

x+40 = (4/3)(3)(4x+1) - 2(4/3)

(4/3)(3) simplifies to just 4, so now we have:

x+40 = 4(4x+1) - 8/3

Mult. all 3 terms by 3 to eliminate the fraction:

3x + 120 = 12(4x+1) - 8

Perform the multiplication:

3x + 120  = 48x + 12 - 8,            or            116 = 45x

Solving for x,    x = 2.578, or     x = 7/90 (answer)

We could also simplify the original equation,  0.75(x+40)=3(4x+1)-2, by multiplying all 3 terms by 100:

75x + 3000 = 1200x + 300 - 200

Then 3000 - 300 + 200 = 1125x, and x = 7/90 (same as before)