Solution - Absolute value equations

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

Break up the fraction:

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

Subtract $$$$ from both sides:

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

Group like terms:

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

Group the coefficients:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

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

Group like terms:

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

Simplify the arithmetic:

$$\frac{-3}{2}x+\frac{3}{2}=-1$$

Subtract $$$$ from both sides:

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

Combine the fractions:

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

Combine the numerators:

$$\frac{-3}{2}x+\frac{0}{2}=-1-\frac{3}{2}$$

Reduce the zero numerator:

$$\frac{-3}{2}x+0=-1-\frac{3}{2}$$

Simplify the arithmetic:

$$\frac{-3}{2}x=-1-\frac{3}{2}$$

Convert the integer into a fraction:

$$\frac{-3}{2}x=\frac{-2}{2}+\frac{-3}{2}$$

Combine the fractions:

$$\frac{-3}{2}x=\frac{\left(-2-3\right)}{2}$$

Combine the numerators:

$$\frac{-3}{2}x=\frac{-5}{2}$$

Multiply both sides by inverse fraction $$$$:

$$\left(\frac{-3}{2}x\right)·\frac{2}{-3}=\left(\frac{-5}{2}\right)·\frac{2}{-3}$$

Move the negative sign from the denominator to the numerator:

$$\frac{-3}{2}x·\frac{-2}{3}=\left(\frac{-5}{2}\right)·\frac{2}{-3}$$

Group like terms:

$$\left(\frac{-3}{2}·\frac{-2}{3}\right)x=\left(\frac{-5}{2}\right)·\frac{2}{-3}$$

Multiply the coefficients:

$$\frac{\left(-3·-2\right)}{\left(2·3\right)}x=\left(\frac{-5}{2}\right)·\frac{2}{-3}$$

Simplify the arithmetic:

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

$$x=\left(\frac{-5}{2}\right)·\frac{2}{-3}$$

Move the negative sign from the denominator to the numerator:

$$x=\frac{-5}{2}·\frac{-2}{3}$$

Multiply the fraction(s):

$$x=\frac{\left(-5·-2\right)}{\left(2·3\right)}$$

Simplify the arithmetic:

$$x=\frac{5}{3}$$

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

Break up the fraction:

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

Expand the parentheses:

$$\frac{x}{2}+\frac{3}{2}=-2x+1$$

Add $$$$ to both sides:

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

Group like terms:

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

Group the coefficients:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

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

Group like terms:

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

Simplify the arithmetic:

$$\frac{5}{2}x+\frac{3}{2}=1$$

Subtract $$$$ from both sides:

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

Combine the fractions:

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

Combine the numerators:

$$\frac{5}{2}x+\frac{0}{2}=1-\frac{3}{2}$$

Reduce the zero numerator:

$$\frac{5}{2}x+0=1-\frac{3}{2}$$

Simplify the arithmetic:

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

Convert the integer into a fraction:

$$\frac{5}{2}x=\frac{2}{2}+\frac{-3}{2}$$

Combine the fractions:

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

Combine the numerators:

$$\frac{5}{2}x=\frac{-1}{2}$$

Multiply both sides by inverse fraction $$$$:

$$\left(\frac{5}{2}x\right)·\frac{2}{5}=\left(\frac{-1}{2}\right)·\frac{2}{5}$$

Group like terms:

$$\left(\frac{5}{2}·\frac{2}{5}\right)x=\left(\frac{-1}{2}\right)·\frac{2}{5}$$

Multiply the coefficients:

$$\frac{\left(5·2\right)}{\left(2·5\right)}x=\left(\frac{-1}{2}\right)·\frac{2}{5}$$

Simplify the fraction:

$$x=\left(\frac{-1}{2}\right)·\frac{2}{5}$$

Multiply the fraction(s):

$$x=\frac{\left(-1·2\right)}{\left(2·5\right)}$$

Simplify the arithmetic:

$$x=\frac{-1}{5}$$