Find All Complex Solutions |x+3|=7x-4

$| x + 3 | = 7 x - 4$

Step 1

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

$x + 3 = \pm (7 x - 4)$

Step 2

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x + 3 = 7 x - 4$

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

Subtract $7 x$ from both sides of the equation.

$x + 3 - 7 x = - 4$

Subtract $7 x$ from $x$ .

$- 6 x + 3 = - 4$

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

Subtract $3$ from both sides of the equation.

$- 6 x = - 4 - 3$

Subtract $3$ from $- 4$ .

$- 6 x = - 7$

Divide each term in $- 6 x = - 7$ by $- 6$ and simplify.

Divide each term in $- 6 x = - 7$ by $- 6$ .

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

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{7}{6}$

Next, use the negative value of the $\pm$ to find the second solution.

$x + 3 = - (7 x - 4)$

Simplify $- (7 x - 4)$ .

Rewrite.

$x + 3 = 0 + 0 - (7 x - 4)$

Simplify by adding zeros.

$x + 3 = - (7 x - 4)$

Apply the distributive property.

$x + 3 = - (7 x) - - 4$

Multiply.

Multiply $7$ by $- 1$ .

$x + 3 = - 7 x - - 4$

Multiply $- 1$ by $- 4$ .

$x + 3 = - 7 x + 4$

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

Add $7 x$ to both sides of the equation.

$x + 3 + 7 x = 4$

Add $x$ and $7 x$ .

$8 x + 3 = 4$

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

Subtract $3$ from both sides of the equation.

$8 x = 4 - 3$

Subtract $3$ from $4$ .

$8 x = 1$

Divide each term in $8 x = 1$ by $8$ and simplify.

Divide each term in $8 x = 1$ by $8$ .

$\frac{8 x}{8} = \frac{1}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{1}{8}$

Divide $x$ by $1$ .

$x = \frac{1}{8}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{7}{6}, \frac{1}{8}$

Step 3

Exclude the solutions that do not make $| x + 3 | = 7 x - 4$ true.

$x = \frac{7}{6}$

Find All Complex Solutions |x+3|=7x-4

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