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

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

Step 1

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

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

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.

$x + 3 = 4 x - 7$

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

Subtract $4 x$ from both sides of the equation.

$x + 3 - 4 x = - 7$

Subtract $4 x$ from $x$ .

$- 3 x + 3 = - 7$

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

Subtract $3$ from both sides of the equation.

$- 3 x = - 7 - 3$

Subtract $3$ from $- 7$ .

$- 3 x = - 10$

Divide each term in $- 3 x = - 10$ by $- 3$ and simplify.

Divide each term in $- 3 x = - 10$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 10}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 10}{- 3}$

Divide $x$ by $1$ .

$x = \frac{- 10}{- 3}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{10}{3}$

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

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

Simplify $- (4 x - 7)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Multiply.

Multiply $4$ by $- 1$ .

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

Multiply $- 1$ by $- 7$ .

$x + 3 = - 4 x + 7$

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

Add $4 x$ to both sides of the equation.

$x + 3 + 4 x = 7$

Add $x$ and $4 x$ .

$5 x + 3 = 7$

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

Subtract $3$ from both sides of the equation.

$5 x = 7 - 3$

Subtract $3$ from $7$ .

$5 x = 4$

Divide each term in $5 x = 4$ by $5$ and simplify.

Divide each term in $5 x = 4$ by $5$ .

$\frac{5 x}{5} = \frac{4}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{4}{5}$

Divide $x$ by $1$ .

$x = \frac{4}{5}$

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

$x = \frac{10}{3}, \frac{4}{5}$

Step 3

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

$x = \frac{10}{3}$

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

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