Solve the following equations and verify the results.
`0.6x+(2x)/3=12x+1.5`

Options

A. `x = (49)/312`
B. `x = (37)/312`
C. `x = (45)/322`
D. `x = (-45)/322`

Answer

To solve the equation \(0.6x + \frac{2x}{3} = 12x + 1.5\) and verify the results, we will follow these steps:

### Step 1: Move all terms involving \(x\) to one side
We start with the equation:
\[ 0.6x + \frac{2x}{3} - 12x = 1.5 \]

### Step 2: Combine like terms
To combine the \(x\) terms, we need a common denominator. The common denominator for \(0.6\) (which is \(\frac{6}{10}\) or \(\frac{3}{5}\)) and \(\frac{2}{3}\) is \(15\). We convert \(0.6\) and \(12\) to fractions with a denominator of \(15\):
\[ 0.6 = \frac{9}{15}, \quad 12 = \frac{180}{15} \]
Now we rewrite the equation:
\[ \frac{9}{15}x + \frac{10}{15}x - \frac{180}{15}x = 1.5 \]

### Step 3: Combine the fractions
Now we can combine the fractions on the left side:
\[ \left(\frac{9 + 10 - 180}{15}\right)x = 1.5 \]
This simplifies to:
\[ \frac{-161}{15}x = 1.5 \]

### Step 4: Isolate \(x\)
To isolate \(x\), we multiply both sides by the reciprocal of \(\frac{-161}{15}\):
\[ x = 1.5 \times \frac{-15}{161} \]
Calculating the right side:
\[ x = \frac{-22.5}{161} \]

### Step 5: Convert to a more manageable fraction
To express this in a simpler form, we can write:
\[ x = \frac{-225}{1610} = \frac{-45}{322} \]

### Step 6: Verification
Now we will verify our solution by substituting \(x = \frac{-45}{322}\) back into the original equation.

**Left Hand Side (LHS):**
\[ 0.6\left(\frac{-45}{322}\right) + \frac{2}{3}\left(\frac{-45}{322}\right) \]
Calculating \(0.6 \times \frac{-45}{322}\):
\[ = \frac{-27}{322} \]
Calculating \(\frac{2}{3} \times \frac{-45}{322}\):
\[ = \frac{-30}{322} \]
Now combining these:
\[ \frac{-27 - 30}{322} = \frac{-57}{322} \]

**Right Hand Side (RHS):**
\[ 12\left(\frac{-45}{322}\right) + 1.5 \]
Calculating \(12 \times \frac{-45}{322}\):
\[ = \frac{-540}{322} \]
Calculating \(1.5\) as a fraction:
\[ 1.5 = \frac{483}{322} \]
Now combining these:
\[ \frac{-540 + 483}{322} = \frac{-57}{322} \]

Since both LHS and RHS are equal:
\[ \frac{-57}{322} = \frac{-57}{322} \]
This confirms that our solution is correct.

### Final Answer:
\[ x = \frac{-45}{322} \]