Question

solve the equation and check.
\\(\frac{2x - 7}{12} + x = \frac{2x + 7}{2} + 1\\)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution is \\(\square\\)
(type an integer or a simplified fraction.)
b. the solution is all real numbers.
c. there is no solution.

Explanation:

Step1: Eliminate denominators

Multiply each term by 12 (the least common multiple of 12 and 2) to eliminate fractions:
$$12\times\frac{2x - 7}{12}+12\times x = 12\times\frac{2x + 7}{2}+12\times1$$
Simplify each term:
$$2x - 7 + 12x = 6(2x + 7)+12$$

Step2: Simplify both sides

Combine like terms on the left:
$$14x - 7 = 12x + 42 + 12$$
Simplify the right side:
$$14x - 7 = 12x + 54$$

Step3: Solve for x

Subtract \(12x\) from both sides:
$$14x - 12x - 7 = 12x - 12x + 54$$
$$2x - 7 = 54$$
Add 7 to both sides:
$$2x - 7 + 7 = 54 + 7$$
$$2x = 61$$
Divide both sides by 2:
$$x=\frac{61}{2}$$

Step4: Check the solution

Substitute \(x = \frac{61}{2}\) back into the original equation:
Left side: \(\frac{2\times\frac{61}{2}-7}{12}+\frac{61}{2}=\frac{61 - 7}{12}+\frac{61}{2}=\frac{54}{12}+\frac{61}{2}=\frac{9}{2}+\frac{61}{2}=\frac{70}{2}=35\)
Right side: \(\frac{2\times\frac{61}{2}+7}{2}+1=\frac{61 + 7}{2}+1=\frac{68}{2}+1 = 34 + 1 = 35\)
Both sides are equal, so the solution is correct.

Answer:

A. The solution is \(\frac{61}{2}\)