Solve for x |x+1|+|3x-2|=2

$| x + 1 | + | 3 x - 2 | = 2$

Step 1

Subtract $| x + 1 |$ from both sides of the equation.

$| 3 x - 2 | = 2 - | x + 1 |$

Step 2

Solve for $| x + 1 |$ .

Rewrite the equation as $2 - | x + 1 | = | 3 x - 2 |$ .

$2 - | x + 1 | = | 3 x - 2 |$

Subtract $2$ from both sides of the equation.

$- | x + 1 | = | 3 x - 2 | - 2$

Divide each term in $- | x + 1 | = | 3 x - 2 | - 2$ by $- 1$ and simplify.

Divide each term in $- | x + 1 | = | 3 x - 2 | - 2$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $| x + 1 |$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{| 3 x - 2 |}{- 1}$ .

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

Rewrite $- 1 \cdot | 3 x - 2 |$ as $- | 3 x - 2 |$ .

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

Divide $- 2$ by $- 1$ .

$| x + 1 | = - | 3 x - 2 | + 2$

Step 3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x + 1 = \pm (- | 3 x - 2 | + 2)$

Step 4

The result consists of both the positive and negative portions of the $\pm$ .

$x + 1 = - | 3 x - 2 | + 2$

$x + 1 = - (- | 3 x - 2 | + 2)$

Step 5

Solve $x + 1 = - | 3 x - 2 | + 2$ for $x$ .

Solve for $| 3 x - 2 |$ .

Rewrite the equation as $- | 3 x - 2 | + 2 = x + 1$ .

$- | 3 x - 2 | + 2 = x + 1$

Move all terms not containing $| 3 x - 2 |$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$- | 3 x - 2 | = x + 1 - 2$

Subtract $2$ from $1$ .

$- | 3 x - 2 | = x - 1$

Divide each term in $- | 3 x - 2 | = x - 1$ by $- 1$ and simplify.

Divide each term in $- | 3 x - 2 | = x - 1$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $| 3 x - 2 |$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{x}{- 1}$ .

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

Rewrite $- 1 \cdot x$ as $- x$ .

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

Divide $- 1$ by $- 1$ .

$| 3 x - 2 | = - x + 1$

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$3 x - 2 = \pm (- x + 1)$

The result consists of both the positive and negative portions of the $\pm$ .

$3 x - 2 = - x + 1$

$3 x - 2 = - (- x + 1)$

Solve $3 x - 2 = - x + 1$ for $x$ .

Move all terms containing $x$ to the left side of the equation.

Add $x$ to both sides of the equation.

$3 x - 2 + x = 1$

Add $3 x$ and $x$ .

$4 x - 2 = 1$

Move all terms not containing $x$ to the right side of the equation.

Add $2$ to both sides of the equation.

$4 x = 1 + 2$

Add $1$ and $2$ .

$4 x = 3$

Divide each term in $4 x = 3$ by $4$ and simplify.

Divide each term in $4 x = 3$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Solve $3 x - 2 = - (- x + 1)$ for $x$ .

Simplify $- (- x + 1)$ .

Rewrite.

$3 x - 2 = 0 + 0 - (- x + 1)$

Simplify by adding zeros.

$3 x - 2 = - (- x + 1)$

Apply the distributive property.

$3 x - 2 = - - x - 1 \cdot 1$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$3 x - 2 = 1 x - 1 \cdot 1$

Multiply $x$ by $1$ .

$3 x - 2 = x - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$3 x - 2 = x - 1$

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$3 x - 2 - x = - 1$

Subtract $x$ from $3 x$ .

$2 x - 2 = - 1$

Move all terms not containing $x$ to the right side of the equation.

Add $2$ to both sides of the equation.

$2 x = - 1 + 2$

Add $- 1$ and $2$ .

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Consolidate the solutions.

$x = \frac{3}{4}, \frac{1}{2}$

Step 6

Solve $x + 1 = - (- | 3 x - 2 | + 2)$ for $x$ .

Solve for $| 3 x - 2 |$ .

Rewrite the equation as $- (- | 3 x - 2 | + 2) = x + 1$ .

$- (- | 3 x - 2 | + 2) = x + 1$

Simplify $- (- | 3 x - 2 | + 2)$ .

Apply the distributive property.

$- - | 3 x - 2 | - 1 \cdot 2 = x + 1$

Multiply $- - | 3 x - 2 |$ .

Multiply $- 1$ by $- 1$ .

$1 | 3 x - 2 | - 1 \cdot 2 = x + 1$

Multiply $| 3 x - 2 |$ by $1$ .

$| 3 x - 2 | - 1 \cdot 2 = x + 1$

Multiply $- 1$ by $2$ .

$| 3 x - 2 | - 2 = x + 1$

Move all terms not containing $| 3 x - 2 |$ to the right side of the equation.

Add $2$ to both sides of the equation.

$| 3 x - 2 | = x + 1 + 2$

Add $1$ and $2$ .

$| 3 x - 2 | = x + 3$

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$3 x - 2 = \pm (x + 3)$

The result consists of both the positive and negative portions of the $\pm$ .

$3 x - 2 = x + 3$

$3 x - 2 = - (x + 3)$

Solve $3 x - 2 = x + 3$ for $x$ .

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$3 x - 2 - x = 3$

Subtract $x$ from $3 x$ .

$2 x - 2 = 3$

Move all terms not containing $x$ to the right side of the equation.

Add $2$ to both sides of the equation.

$2 x = 3 + 2$

Add $3$ and $2$ .

$2 x = 5$

Divide each term in $2 x = 5$ by $2$ and simplify.

Divide each term in $2 x = 5$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Solve $3 x - 2 = - (x + 3)$ for $x$ .

Simplify $- (x + 3)$ .

Rewrite.

$3 x - 2 = 0 + 0 - (x + 3)$

Simplify by adding zeros.

$3 x - 2 = - (x + 3)$

Apply the distributive property.

$3 x - 2 = - x - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$3 x - 2 = - x - 3$

Move all terms containing $x$ to the left side of the equation.

Add $x$ to both sides of the equation.

$3 x - 2 + x = - 3$

Add $3 x$ and $x$ .

$4 x - 2 = - 3$

Move all terms not containing $x$ to the right side of the equation.

Add $2$ to both sides of the equation.

$4 x = - 3 + 2$

Add $- 3$ and $2$ .

$4 x = - 1$

Divide each term in $4 x = - 1$ by $4$ and simplify.

Divide each term in $4 x = - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4}$

Divide $x$ by $1$ .

$x = \frac{- 1}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

Consolidate the solutions.

$x = \frac{5}{2}, - \frac{1}{4}$

Step 7

Consolidate the solutions.

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

Step 8

Exclude the solutions that do not make $| x + 1 | + | 3 x - 2 | = 2$ true.

$x = \frac{3}{4}, \frac{1}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{4}, \frac{1}{2}$

Decimal Form:

$x = 0.75, 0.5$

Step 10

Solve for x |x+1|+|3x-2|=2

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