Solve and check the equation.

Question

2(5 - 3x) = 3(2x + 1)
Select the correct choice below​ and, if​ necessary, fill in any answer box to complete your answer.
A. The solution set is { }.
B. The solution set is {x| x is a real number}.
C. The solution set is empty set

Answer 1

To solve the equation \(2(5 - 3x) = 3(2x + 1)\), let's start by expanding both sides.

1. Distribute on the left side: \[ 2(5 - 3x) = 10 - 6x \]

2. Distribute on the right side: \[ 3(2x + 1) = 6x + 3 \]

Now the equation becomes: \[ 10 - 6x = 6x + 3 \]

3. Next, let's move all terms involving \(x\) to one side and constant terms to the other side. Start by adding \(6x\) to both sides: \[ 10 = 12x + 3 \]

4. Now, subtract \(3\) from both sides: \[ 7 = 12x \]

5. Finally, divide by \(12\): \[ x = \frac{7}{12} \]

Now that we have the solution \(x = \frac{7}{12}\), let's check it by substituting it back into the original equation.

Original equation: \[ 2(5 - 3x) = 3(2x + 1) \] Substituting \(x = \frac{7}{12}\):

Calculate the left side: \[ 2(5 - 3 \cdot \frac{7}{12}) = 2(5 - \frac{21}{12}) = 2(5 - 1.75) = 2(3.25) = 6.5 \]

Calculate the right side: \[ 3(2 \cdot \frac{7}{12} + 1) = 3(\frac{14}{12} + 1) = 3(\frac{14}{12} + \frac{12}{12}) = 3(\frac{26}{12}) = 3 \cdot \frac{13}{6} = \frac{39}{6} = 6.5 \]

Both sides are equal, so the solution is verified.

Hence, the correct choice is: A. The solution set is \({ \frac{7}{12} }\).