Question

which of the following best describes the solution to the equation below?
30x + 27 = -45x - 6
a. exactly one real solution, ( x = 2\frac{1}{5} )
b. infinite real solutions
c. no real solutions
d. exactly one real solution, ( x = -\frac{11}{25} )

Explanation:

Step1: Combine like terms (x terms)

Add \(45x\) to both sides of the equation \(30x + 27=-45x - 6\) to get \(30x + 45x+27=-45x + 45x-6\), which simplifies to \(75x + 27=-6\).

Step2: Isolate the x term

Subtract 27 from both sides: \(75x+27 - 27=-6 - 27\), so \(75x=-33\).

Step3: Solve for x

Divide both sides by 75: \(x=\frac{-33}{75}\), simplify the fraction by dividing numerator and denominator by 3, we get \(x =-\frac{11}{25}\). Since we found a unique value for \(x\), there is exactly one real solution.

Answer:

D. exactly one real solution, \(x = -\frac{11}{25}\)