Question

x/3 + 3/2 < x/4 + 2, x€N

Answer 1

Answer:

How can one solve the following by using the method of factorization: 4/(x-3) = 5/(2x +3) where x ≠ 3 and x ≠ -3/2?

The last two statements just make sure that we will not be dividing by 0, we can ignore them until we have solutions.

The concept of solving by factorization usually applies to quadratics and greater, but this becomes a linear equation..

4/(x-3) = 5/(2x +3) : Multiply both sides by the common denominator (x-3)(2x+3)

4/(x-3) • (x-3)(2x+3)= 5/(2x +3) • (x-3)(2x+3) : Cancel the common factors

4(2x +3) = 5(x-3) : Simplify

8x + 12 = 5x - 15 : Subtract 5x from both sides and subtract 12 from both sides so that you move the variables to one side the constants to the other.

8x - 5x + 12 - 12 = 5x - 5x - 15 - 12 : Simplify

3x = -27 : Divide both sides by 3 to isolate the variable

x = -9

Now consider the prohibitions on x given, x ≠ 3 and x ≠ -3/2, neither of these are violated when x = -9

Check: 4/(x-3) = 5/(2x +3)

4/(-9-3) = 5/(2(-9) +3)

4/-12 = 5/(-18 + 3)

1/-3 = 5/(-15)

-(1/3) = -(1/3)