Sum of all the solutions of the equation `|x-3|+|x+5|=7x,` is :

Options

A. `6/7`
B. `8/7`
C. `58/63`
D. `8/45`

Answer

To solve the equation \( |x-3| + |x+5| = 7x \), we will analyze the equation by considering different intervals based on the critical points where the absolute values change, which are \( x = 3 \) and \( x = -5 \).

### Step 1: Identify the intervals
The critical points divide the number line into three intervals:
1. \( x < -5 \)
2. \( -5 \leq x < 3 \)
3. \( x \geq 3 \)

### Step 2: Solve for each interval

#### Interval 1: \( x < -5 \)
In this interval, both expressions inside the absolute values are negative:
\[ |x-3| = 3 - x \quad \text{and} \quad |x+5| = - (x + 5) = -x - 5 \]
Substituting these into the equation gives:
\[ (3 - x) + (-x - 5) = 7x \]
Simplifying this:
\[ 3 - x - x - 5 = 7x \implies -2 - 2x = 7x \implies -2 = 9x \implies x = -\frac{2}{9} \]
Since \( -\frac{2}{9} \) is greater than \(-5\), we reject this solution.

#### Interval 2: \( -5 \leq x < 3 \)
In this interval, the expression for \( |x-3| \) is negative and \( |x+5| \) is positive:
\[ |x-3| = 3 - x \quad \text{and} \quad |x+5| = x + 5 \]
Substituting these into the equation gives:
\[ (3 - x) + (x + 5) = 7x \]
Simplifying this:
\[ 3 - x + x + 5 = 7x \implies 8 = 7x \implies x = \frac{8}{7} \]
Since \( \frac{8}{7} \) is between \(-5\) and \(3\), we accept this solution.

#### Interval 3: \( x \geq 3 \)
In this interval, both expressions inside the absolute values are positive:
\[ |x-3| = x - 3 \quad \text{and} \quad |x+5| = x + 5 \]
Substituting these into the equation gives:
\[ (x - 3) + (x + 5) = 7x \]
Simplifying this:
\[ x - 3 + x + 5 = 7x \implies 2 + 2x = 7x \implies 2 = 5x \implies x = \frac{2}{5} \]
Since \( \frac{2}{5} \) is less than \(3\), we reject this solution.

### Step 3: Sum of all solutions
The only valid solution we found is \( x = \frac{8}{7} \). Therefore, the sum of all solutions is:
\[ \text{Sum of all solutions} = \frac{8}{7} \]

### Final Answer
The sum of all the solutions of the equation \( |x-3| + |x+5| = 7x \) is \( \frac{8}{7} \).

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