Question

Kelly tells you that when variables are in the denominator in the equation 1/2+3/x=3/4 becomes unsolvable. Kelly explains, "There is a value for x that makes the denominator zero, and you can't divide by zero." Demonstrate to Kelly how the equation is still solvable and explain your reason

Answer 1

Kelly is quite right that the denominator can not be zero. So let's not forget about that as we go through our steps.


`\rm \frac12+\frac3x=\frac34`

To get rid of the fractions let's multiply both sides of the equation by the Least Common Multiple of our denominators. We see a 2, 4 and x. The LCM of these values is 4x so we'll multiply both sides by 4x.


`\rm 4x\left(\frac12+\frac3x\right)=\left(\frac34\right)4x`

distribute the 4x to each term on the left,
and cancel stuff out as needed,
and do the same on the right side of the equation,


`\rm 2x+12=3x`

We've taken x out of the denominator though. So we should keep a note somewhere on the side of the page that x can not equal zero still. Even though we moved some things around, make our equation look different, we still have that restriction.


`\rm 2x+12=3x\qquad\qquad\qquad x\\e0`

From this point, solve for x using Algebra:
Subtract 2x from each side,


`\rm 12=x`

12 is a solution, and it is not 0.
If the solution had worked out to be x=0, we would reject it.