Question

How do you solve `\frac{1}{4} - x = \frac{1}{2} x + \frac{9}{4}`?

Answer 1

`x=-4/3`

Explanation:

Multiply by 1 and you do not change the value. However, 1 comes in many forms.

The objective is to make all the denominators the same and then discard them. We can do that and I will show you why.

Write as:

`1/4 -(x color(red)(xx 1))=(1/2x color(green)(xx 1)) +9/4`...........Equation(1)

`1/4-(x color(red)(xx 4/4))=(1/2xcolor(green)( xx 2/2)) +9/4`

`1/4-(4x)/4" "=" "(2x)/4+9/4` ......................Equation(2)
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Mathematically we can multiply both sides by 4. This gets rid of the denominator. Let me show you:

Write equation(2) as:

`1/4(1-4x) = 1/4(2x+9)`

Multiply both sides by 4 giving:

`4/4(1-4x)=4/4(2x+9)`

But `4/4=1`

`1-4x=2x+9`
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Add `4x` to both sides

`1=6x+9`

Subtract 9 from both sides

`-8=6x`

Divide both sides by 6

`-8/6=x`

`x= - (8-:2)/(6-:2) = -4/3`