Solution - Absolute value equations

$$\left(\frac{-1}{2}n+7\right)-\frac{5}{3}·n=\left(\frac{5}{3}n\right)-\frac{5}{3}n$$

Group like terms:

$$\left(\frac{-1}{2}·n+\frac{-5}{3}·n\right)+7=\left(\frac{5}{3}·n\right)-\frac{5}{3}n$$

Group the coefficients:

$$\left(\frac{-1}{2}+\frac{-5}{3}\right)n+7=\left(\frac{5}{3}·n\right)-\frac{5}{3}n$$

Find the lowest common denominator:

$$\left(\frac{\left(-1·3\right)}{\left(2·3\right)}+\frac{\left(-5·2\right)}{\left(3·2\right)}\right)n+7=\left(\frac{5}{3}·n\right)-\frac{5}{3}n$$

Multiply the denominators:

$$\left(\frac{\left(-1·3\right)}{6}+\frac{\left(-5·2\right)}{6}\right)n+7=\left(\frac{5}{3}·n\right)-\frac{5}{3}n$$

Multiply the numerators:

$$\left(\frac{-3}{6}+\frac{-10}{6}\right)n+7=\left(\frac{5}{3}·n\right)-\frac{5}{3}n$$

Combine the fractions:

$$\frac{\left(-3-10\right)}{6}·n+7=\left(\frac{5}{3}·n\right)-\frac{5}{3}n$$

Combine the numerators:

$$\frac{-13}{6}·n+7=\left(\frac{5}{3}·n\right)-\frac{5}{3}n$$

Combine the fractions:

$$\frac{-13}{6}·n+7=\frac{\left(5-5\right)}{3}n$$

Combine the numerators:

$$\frac{-13}{6}·n+7=\frac{0}{3}n$$

Reduce the zero numerator:

$$\frac{-13}{6}n+7=0n$$

Simplify the arithmetic:

$$\frac{-13}{6}n+7=0$$

Subtract $$$$ from both sides:

$$\left(\frac{-13}{6}n+7\right)-7=0-7$$

Simplify the arithmetic:

$$\frac{-13}{6}n=0-7$$

Simplify the arithmetic:

$$\frac{-13}{6}n=-7$$

Multiply both sides by inverse fraction $$$$:

$$\left(\frac{-13}{6}n\right)·\frac{6}{-13}=-7·\frac{6}{-13}$$

Move the negative sign from the denominator to the numerator:

$$\frac{-13}{6}n·\frac{-6}{13}=-7·\frac{6}{-13}$$

Group like terms:

$$\left(\frac{-13}{6}·\frac{-6}{13}\right)n=-7·\frac{6}{-13}$$

Multiply the coefficients:

$$\frac{\left(-13·-6\right)}{\left(6·13\right)}n=-7·\frac{6}{-13}$$

Simplify the arithmetic:

$$1n=-7·\frac{6}{-13}$$

$$n=-7·\frac{6}{-13}$$

Move the negative sign from the denominator to the numerator:

$$n=-7·\frac{-6}{13}$$

Multiply the fraction(s):

$$n=\frac{\left(-7·-6\right)}{13}$$

Simplify the arithmetic:

$$n=\frac{42}{13}$$

$$\left(\frac{-1}{2}n+7\right)-7=\left(\frac{-5}{3}n\right)-7$$

Simplify the arithmetic:

$$\frac{-1}{2}·n=\left(\frac{-5}{3}n\right)-7$$

Add $$$$ to both sides:

$$\left(\frac{-1}{2}n\right)+\frac{5}{3}·n=\left(\frac{-5}{3}n-7\right)+\frac{5}{3}n$$

Group the coefficients:

$$\left(\frac{-1}{2}+\frac{5}{3}\right)n=\left(\frac{-5}{3}·n-7\right)+\frac{5}{3}n$$

Find the lowest common denominator:

$$\left(\frac{\left(-1·3\right)}{\left(2·3\right)}+\frac{\left(5·2\right)}{\left(3·2\right)}\right)n=\left(\frac{-5}{3}·n-7\right)+\frac{5}{3}n$$

Multiply the denominators:

$$\left(\frac{\left(-1·3\right)}{6}+\frac{\left(5·2\right)}{6}\right)n=\left(\frac{-5}{3}·n-7\right)+\frac{5}{3}n$$

Multiply the numerators:

$$\left(\frac{-3}{6}+\frac{10}{6}\right)n=\left(\frac{-5}{3}·n-7\right)+\frac{5}{3}n$$

Combine the fractions:

$$\frac{\left(-3+10\right)}{6}·n=\left(\frac{-5}{3}·n-7\right)+\frac{5}{3}n$$

Combine the numerators:

$$\frac{7}{6}·n=\left(\frac{-5}{3}·n-7\right)+\frac{5}{3}n$$

Group like terms:

$$\frac{7}{6}·n=\left(\frac{-5}{3}·n+\frac{5}{3}n\right)-7$$

Combine the fractions:

$$\frac{7}{6}·n=\frac{\left(-5+5\right)}{3}n-7$$

Combine the numerators:

$$\frac{7}{6}·n=\frac{0}{3}n-7$$

Reduce the zero numerator:

$$\frac{7}{6}n=0n-7$$

Simplify the arithmetic:

$$\frac{7}{6}n=-7$$

Multiply both sides by inverse fraction $$$$:

$$\left(\frac{7}{6}n\right)·\frac{6}{7}=-7·\frac{6}{7}$$

Group like terms:

$$\left(\frac{7}{6}·\frac{6}{7}\right)n=-7·\frac{6}{7}$$

Multiply the coefficients:

$$\frac{\left(7·6\right)}{\left(6·7\right)}n=-7·\frac{6}{7}$$

Simplify the fraction:

$$n=-7·\frac{6}{7}$$

Multiply the fraction(s):

$$n=\frac{\left(-7·6\right)}{7}$$

Simplify the arithmetic:

$$n=-6$$