Question

solve for x: 1/3x+1/2=-2/3x+2/5

Answer 1

1) multiply both sides by 3x:\[3x(\frac{ 1 }{ 3x }+\frac{ 1 }{ 2 })=(\frac{ -2 }{ 3x }+\frac{ 2 }{ 5 })\] to get \[\frac{ 3x }{ 2 }+1=\frac{ 6x }{ 5 }-2\]
Getting a common denominator:
\[\frac{ 5 }{ 5 }(\frac{ 3x }{ 2 }+1)=\frac{ 2 }{ 2 }(\frac{ 6x }{ 5 }-2)\]
This is allowed since 5/5 and 2/2 are actually just 1, so they don't affect the equation at all. So now we get \[\frac{ 15x }{ 10 }+1=\frac{ 12x }{ 10 }-2\]
Reducing:\[\frac{ 3x }{ 10 }=-3\]
Giving us: \[3x=-30\]
and \[x=-10\]

Answer 2

find the common denominator of all four denominators (3,2,3,5) which is 30.
Multiply the equation by 30 to get rid of the fractions...
(30)1/3x + (30)1/2= (30)-2/3x + (30)2/5
10x + 15 = -20x + 12 (now get x's on one side, everything else on the other )
10x + 20x = 12 - 15
30x = - 3 (divide by 30 to get x by itself)
30x/30 = - 3/30
x = - 1/10

check...
10x + 15 = - 20x + 12
10(-1/10) + 15 = - 20(-1/10) + 12
-1 + 15 = 2 + 12
14 = 14 (correct)