If `(x + 3)/(2) + 9x = 5(x - 3) + (x + 23)/(5)` then what is the value of x?

Options

A. 7
B. 8
C. 10
D. 12

Answer

To solve the equation \(\frac{x + 3}{2} + 9x = 5(x - 3) + \frac{x + 23}{5}\), we will follow these steps:

### Step 1: Eliminate the fractions
To eliminate the fractions, we can find a common denominator. The denominators in the equation are 2 and 5. The least common multiple (LCM) of 2 and 5 is 10. We will multiply every term in the equation by 10.

\[ 10 \left(\frac{x + 3}{2}\right) + 10(9x) = 10\left(5(x - 3)\right) + 10\left(\frac{x + 23}{5}\right) \]

### Step 2: Simplify the equation
Now, let's simplify each term:

\[ 5(x + 3) + 90x = 50(x - 3) + 2(x + 23) \]

### Step 3: Distribute the terms
Distributing the terms on both sides gives us:

\[ 5x + 15 + 90x = 50x - 150 + 2x + 46 \]

### Step 4: Combine like terms
Now, combine like terms on both sides:

Left side:
\[ 5x + 90x = 95x \implies 95x + 15 \]

Right side:
\[ 50x + 2x - 150 + 46 = 52x - 104 \]

So, the equation now looks like:

\[ 95x + 15 = 52x - 104 \]

### Step 5: Move all x terms to one side
Next, we will move all terms involving \(x\) to one side and constant terms to the other side:

\[ 95x - 52x = -104 - 15 \]

### Step 6: Simplify both sides
This simplifies to:

\[ 43x = -119 \]

### Step 7: Solve for x
Now, divide both sides by 43 to isolate \(x\):

\[ x = \frac{-119}{43} \]

Thus, the value of \(x\) is:

\[ x = -\frac{119}{43} \]

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