The solution of the equation ((x + 1) - (2x + 4)) / (3 - 5x) = 1/23 is also…

Concept: When an equation has the form (linear expression in x) / (linear expression in x) = a constant, clear the fraction by cross-multiplying, then collect like terms to isolate x. Once the value of x that satisfies the given equation is known, the same value must also satisfy whichever option equation is being tested; substitute it into each candidate equation and check whether both sides come out equal.

1. Simplify the numerator of the given equation: (x + 1) - (2x + 4) = x + 1 - 2x - 4 = -x - 3, so the equation becomes (-x - 3) / (3 - 5x) = 1/23.

2. Cross-multiply to remove the fraction: 23(-x - 3) = 1(3 - 5x), which gives -23x - 69 = 3 - 5x.

3. Collect the x-terms on one side: -23x + 5x = 3 + 69, so -18x = 72.

4. Divide both sides by -18: x = 72 / (-18) = -4.

5. Substitute x = -4 into 3(2x + 3) = 5(x + 1): the left side gives 3(2(-4) + 3) = 3(-5) = -15, and the right side gives 5(-4 + 1) = 5(-3) = -15. Both sides equal -15, so this equation is satisfied by the same value of x.

Cross-check: Substituting x = -4 back into the original equation confirms it directly — the numerator becomes -(-4) - 3 = 1 and the denominator becomes 3 - 5(-4) = 23, giving 1/23, which matches the right-hand side of the given equation. This confirms x = -4 is correct and that 3(2x + 3) = 5(x + 1) shares this same solution.