Question 444212
1. {{{x/2 = (x+1)/4}}}
Find the least common denominator (LCD) between 2 and 4... 4. Multiply the entire equation by this LCD: {{{4(x/2) = ((x+1)/4)4}}}
Simplify: {{{2x = x + 1}}}
Solve for x by subtracting x from both sides: {{{x = 1}}}
Therefore, x = 1
2. {{{3/(x-1) = 4/(3x + 2)}}}
Find the LCD between x-1 and 3x+2... (x-1)(3x+2) Multiply the entire equation by this LCD: {{{(x-1)(3x+2)(3/(x-1)) = (x-1)(3x+2)(4/(3x+2))}}}
Simplify (NOTE: the (x-1) and /(x-1) cancel each other out, and the same with the (3x+2) and /(3x+2)): {{{3(3x+2) = 4(x-1)}}}
Distribute: {{{9x+6 = 4x-4}}}
Subtract 6 from both sides: {{{9x = 4x-10}}}
Subtract 4x from both sides: {{{5x = -10}}}
Divide both sides by 5: {{{x = -2}}}
Therefore, x = -2
3. {{{3x/(x+1) = 0}}}
Since there is only one denominator, the LCD is x+1.
{{{(x+1)(3x/(x+1)) = 0(x+1)}}}
Simplify (NOTE: The (x+1) and /(x+1) cancel each other out, and 0 multiplied by anything is 0): {{{3x = 0}}}
Divide both sides by 3: {{{x = 0}}}
Therefore, x = 0
4. {{{3/(x-1) = 1/(x^2 - 1)}}}
Factor the denominator ({{{x^2 - 1}}} is a difference of squares, so (x+1)(x-1)): {{{3/(x-1) = 1/(x+1)(x-1)}}}
Find the LCD between x-1 and (x+1)(x-1)... (x+1)(x-1). Multiply the entire equation by this: {{{(x+1)(x-1)(3/(x-1)) = (x+1)(x-1)(1/(x+1)(x-1))}}}
Simplify: {{{3(x+1) = 1}}}
Distribute: {{{3x + 3 = 1}}}
Subtract 3 from both sides: {{{3x = -2}}}
Divide both sides by 3: {{{x = -2/3}}}
Therefore, {{{x = -2/3}}}