Solve 1/2left(3x-1/2right)-1/3left(4x-1/3right)=frac{1}{4}left(6x-5right)-frac{2}{3}

$$\frac{ 1 }{ 2 } \left( 3x- \frac{ 1 }{ 2 } \right) - \frac{ 1 }{ 3 } \left( 4x- \frac{ 1 }{ 3 } \right) = \frac{ 1 }{ 4 } \left( 6x-5 \right) - \frac{ 2 }{ 3 }$$

Solve for x

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{2} by 3x-\frac{1}{2}.

$$\frac{1}{2}\times 3x+\frac{1}{2}\left(-\frac{1}{2}\right)-\frac{1}{3}\left(4x-\frac{1}{3}\right)=\frac{1}{4}\left(6x-5\right)-\frac{2}{3}$$

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

$$\frac{3}{2}x+\frac{1}{2}\left(-\frac{1}{2}\right)-\frac{1}{3}\left(4x-\frac{1}{3}\right)=\frac{1}{4}\left(6x-5\right)-\frac{2}{3}$$

Multiply \frac{1}{2} times -\frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3}{2}x+\frac{1\left(-1\right)}{2\times 2}-\frac{1}{3}\left(4x-\frac{1}{3}\right)=\frac{1}{4}\left(6x-5\right)-\frac{2}{3}$$

Do the multiplications in the fraction \frac{1\left(-1\right)}{2\times 2}.

$$\frac{3}{2}x+\frac{-1}{4}-\frac{1}{3}\left(4x-\frac{1}{3}\right)=\frac{1}{4}\left(6x-5\right)-\frac{2}{3}$$

Fraction \frac{-1}{4} can be rewritten as -\frac{1}{4} by extracting the negative sign.

$$\frac{3}{2}x-\frac{1}{4}-\frac{1}{3}\left(4x-\frac{1}{3}\right)=\frac{1}{4}\left(6x-5\right)-\frac{2}{3}$$

Use the distributive property to multiply -\frac{1}{3} by 4x-\frac{1}{3}.

$$\frac{3}{2}x-\frac{1}{4}-\frac{1}{3}\times 4x-\frac{1}{3}\left(-\frac{1}{3}\right)=\frac{1}{4}\left(6x-5\right)-\frac{2}{3}$$

Express -\frac{1}{3}\times 4 as a single fraction.

$$\frac{3}{2}x-\frac{1}{4}+\frac{-4}{3}x-\frac{1}{3}\left(-\frac{1}{3}\right)=\frac{1}{4}\left(6x-5\right)-\frac{2}{3}$$

Fraction \frac{-4}{3} can be rewritten as -\frac{4}{3} by extracting the negative sign.

$$\frac{3}{2}x-\frac{1}{4}-\frac{4}{3}x-\frac{1}{3}\left(-\frac{1}{3}\right)=\frac{1}{4}\left(6x-5\right)-\frac{2}{3}$$

Multiply -\frac{1}{3} times -\frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3}{2}x-\frac{1}{4}-\frac{4}{3}x+\frac{-\left(-1\right)}{3\times 3}=\frac{1}{4}\left(6x-5\right)-\frac{2}{3}$$

Do the multiplications in the fraction \frac{-\left(-1\right)}{3\times 3}.

$$\frac{3}{2}x-\frac{1}{4}-\frac{4}{3}x+\frac{1}{9}=\frac{1}{4}\left(6x-5\right)-\frac{2}{3}$$

Combine \frac{3}{2}x and -\frac{4}{3}x to get \frac{1}{6}x.

$$\frac{1}{6}x-\frac{1}{4}+\frac{1}{9}=\frac{1}{4}\left(6x-5\right)-\frac{2}{3}$$

Least common multiple of 4 and 9 is 36. Convert -\frac{1}{4} and \frac{1}{9} to fractions with denominator 36.

$$\frac{1}{6}x-\frac{9}{36}+\frac{4}{36}=\frac{1}{4}\left(6x-5\right)-\frac{2}{3}$$

Since -\frac{9}{36} and \frac{4}{36} have the same denominator, add them by adding their numerators.

$$\frac{1}{6}x+\frac{-9+4}{36}=\frac{1}{4}\left(6x-5\right)-\frac{2}{3}$$

Add -9 and 4 to get -5.

$$\frac{1}{6}x-\frac{5}{36}=\frac{1}{4}\left(6x-5\right)-\frac{2}{3}$$

Use the distributive property to multiply \frac{1}{4} by 6x-5.

$$\frac{1}{6}x-\frac{5}{36}=\frac{1}{4}\times 6x+\frac{1}{4}\left(-5\right)-\frac{2}{3}$$

Multiply \frac{1}{4} and 6 to get \frac{6}{4}.

$$\frac{1}{6}x-\frac{5}{36}=\frac{6}{4}x+\frac{1}{4}\left(-5\right)-\frac{2}{3}$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

$$\frac{1}{6}x-\frac{5}{36}=\frac{3}{2}x+\frac{1}{4}\left(-5\right)-\frac{2}{3}$$

Multiply \frac{1}{4} and -5 to get \frac{-5}{4}.

$$\frac{1}{6}x-\frac{5}{36}=\frac{3}{2}x+\frac{-5}{4}-\frac{2}{3}$$

Fraction \frac{-5}{4} can be rewritten as -\frac{5}{4} by extracting the negative sign.

$$\frac{1}{6}x-\frac{5}{36}=\frac{3}{2}x-\frac{5}{4}-\frac{2}{3}$$

Least common multiple of 4 and 3 is 12. Convert -\frac{5}{4} and \frac{2}{3} to fractions with denominator 12.

$$\frac{1}{6}x-\frac{5}{36}=\frac{3}{2}x-\frac{15}{12}-\frac{8}{12}$$

Since -\frac{15}{12} and \frac{8}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{6}x-\frac{5}{36}=\frac{3}{2}x+\frac{-15-8}{12}$$

Subtract 8 from -15 to get -23.

$$\frac{1}{6}x-\frac{5}{36}=\frac{3}{2}x-\frac{23}{12}$$

Subtract \frac{3}{2}x from both sides.

$$\frac{1}{6}x-\frac{5}{36}-\frac{3}{2}x=-\frac{23}{12}$$

Combine \frac{1}{6}x and -\frac{3}{2}x to get -\frac{4}{3}x.

$$-\frac{4}{3}x-\frac{5}{36}=-\frac{23}{12}$$

Add \frac{5}{36} to both sides.

$$-\frac{4}{3}x=-\frac{23}{12}+\frac{5}{36}$$

Least common multiple of 12 and 36 is 36. Convert -\frac{23}{12} and \frac{5}{36} to fractions with denominator 36.

$$-\frac{4}{3}x=-\frac{69}{36}+\frac{5}{36}$$

Since -\frac{69}{36} and \frac{5}{36} have the same denominator, add them by adding their numerators.

$$-\frac{4}{3}x=\frac{-69+5}{36}$$

Add -69 and 5 to get -64.

$$-\frac{4}{3}x=\frac{-64}{36}$$

Reduce the fraction \frac{-64}{36} to lowest terms by extracting and canceling out 4.

$$-\frac{4}{3}x=-\frac{16}{9}$$

Multiply both sides by -\frac{3}{4}, the reciprocal of -\frac{4}{3}.

$$x=-\frac{16}{9}\left(-\frac{3}{4}\right)$$

Multiply -\frac{16}{9} times -\frac{3}{4} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{-16\left(-3\right)}{9\times 4}$$

Do the multiplications in the fraction \frac{-16\left(-3\right)}{9\times 4}.

$$x=\frac{48}{36}$$

Reduce the fraction \frac{48}{36} to lowest terms by extracting and canceling out 12.

$$x=\frac{4}{3}$$