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

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

Step 1

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

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

Step 2

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

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

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

Step 3

Rewrite the absolute value equation as four equations without absolute value bars.

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

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

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

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

Step 4

After simplifying, there are only two unique equations to be solved.

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

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

Step 5

Solve $3 - x = \frac{3 x - 1}{2}$ for $x$ .

Multiply both sides by $2$ .

$(3 - x) \cdot 2 = \frac{3 x - 1}{2} \cdot 2$

Simplify.

Simplify the left side.

Simplify $(3 - x) \cdot 2$ .

Apply the distributive property.

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

Simplify the expression.

Multiply $3$ by $2$ .

$6 - x \cdot 2 = \frac{3 x - 1}{2} \cdot 2$

Multiply $2$ by $- 1$ .

$6 - 2 x = \frac{3 x - 1}{2} \cdot 2$

Reorder $6$ and $- 2 x$ .

$- 2 x + 6 = \frac{3 x - 1}{2} \cdot 2$

Simplify the right side.

Cancel the common factor of $2$ .

Cancel the common factor.

$- 2 x + 6 = \frac{3 x - 1}{2} \cdot 2$

Rewrite the expression.

$- 2 x + 6 = 3 x - 1$

Solve for $x$ .

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

Subtract $3 x$ from both sides of the equation.

$- 2 x + 6 - 3 x = - 1$

Subtract $3 x$ from $- 2 x$ .

$- 5 x + 6 = - 1$

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

Subtract $6$ from both sides of the equation.

$- 5 x = - 1 - 6$

Subtract $6$ from $- 1$ .

$- 5 x = - 7$

Divide each term in $- 5 x = - 7$ by $- 5$ and simplify.

Divide each term in $- 5 x = - 7$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 7}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 7}{- 5}$

Divide $x$ by $1$ .

$x = \frac{- 7}{- 5}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{7}{5}$

Step 6

Solve $3 - x = - \frac{3 x - 1}{2}$ for $x$ .

Multiply both sides by $2$ .

$(3 - x) \cdot 2 = - \frac{3 x - 1}{2} \cdot 2$

Simplify.

Simplify the left side.

Simplify $(3 - x) \cdot 2$ .

Apply the distributive property.

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

Simplify the expression.

Multiply $3$ by $2$ .

$6 - x \cdot 2 = - \frac{3 x - 1}{2} \cdot 2$

Multiply $2$ by $- 1$ .

$6 - 2 x = - \frac{3 x - 1}{2} \cdot 2$

Reorder $6$ and $- 2 x$ .

$- 2 x + 6 = - \frac{3 x - 1}{2} \cdot 2$

Simplify the right side.

Simplify $- \frac{3 x - 1}{2} \cdot 2$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3 x - 1}{2}$ into the numerator.

$- 2 x + 6 = \frac{- (3 x - 1)}{2} \cdot 2$

Cancel the common factor.

$- 2 x + 6 = \frac{- (3 x - 1)}{2} \cdot 2$

Rewrite the expression.

$- 2 x + 6 = - (3 x - 1)$

Apply the distributive property.

$- 2 x + 6 = - (3 x) - - 1$

Multiply.

Multiply $3$ by $- 1$ .

$- 2 x + 6 = - 3 x - - 1$

Multiply $- 1$ by $- 1$ .

$- 2 x + 6 = - 3 x + 1$

Solve for $x$ .

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

Add $3 x$ to both sides of the equation.

$- 2 x + 6 + 3 x = 1$

Add $- 2 x$ and $3 x$ .

$x + 6 = 1$

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

Subtract $6$ from both sides of the equation.

$x = 1 - 6$

Subtract $6$ from $1$ .

$x = - 5$

Step 7

List all of the solutions.

$x = \frac{7}{5}, - 5$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{5}, - 5$

Decimal Form:

$x = 1.4, - 5$

Mixed Number Form:

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

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

Related Questions

Support LCMGCF.com by disabling your adblocker.