Question

How do you solve `\frac { x } { 2} = \frac { x } { 3} - \frac { 1} { 4}`?

Answer 1

`x=-3/2`

Explanation:

Solve: `x/2=x/3-1/4`

When adding or subtracting fractions, they must have the same denominator, called the least common denominator or LCD. The LCD can be found by listing the multiples of each denominator. The first multiple to be found in the list of multiples for all denominators is the LCD.

`2: 2,4,6,8,10,color(teal)(12),14,16,18,20`
`3: 3,6,9,color(teal)12`
`4: 4,8,color(teal)12`

The LCD is `12`.

Now each fraction must be multiplied by an `color(darkmagenta)"equivalent fraction"` that when multiplied by the original fraction will make the LCD the denominator for each fraction.

`x/2xxcolor(darkmagenta)(6/6)=(6x)/(12)`

`x/3xxcolor(darkmagenta)(4/4)=(4x)/(12)`

`1/4xxcolor(darkmagenta)(3/3)=3/12`

Rewrite the problem.

`(6x)/12=(4x)/(12)-3/12`

Subtract `(4x)/12` from both sides.

`(6x)/12-(4x)/12=-3/12`

Simplify.

`(2x)/12=(-3)/12`

Divide the left side by `2/12`. (Remember when you divide by a fraction, you invert it and multiply.)

`x=(-3)/cancel(12)xxcancel(12)/2`

`x=-3/2`