Question

how many solutions exist for the given equation?
12x + 1 = 3(4x + 1) - 2
zero
one
two
infinitely many

Explanation:

Step1: Expand the right side

First, we expand the right - hand side of the equation \(12x + 1=3(4x + 1)-2\). Using the distributive property \(a(b + c)=ab+ac\), we have \(3(4x + 1)=3\times4x+3\times1 = 12x+3\). So the equation becomes \(12x + 1=12x+3 - 2\).

Step2: Simplify the right side

Simplify the right - hand side: \(12x+3 - 2=12x + 1\). Now our equation is \(12x + 1=12x+1\).

Step3: Analyze the equation

Subtract \(12x\) from both sides of the equation \(12x + 1=12x+1\). We get \((12x + 1)-12x=(12x + 1)-12x\), which simplifies to \(1 = 1\). This is a true statement for all values of \(x\). When we end up with a statement like \(a=a\) (where \(a\) is a non - variable constant) after simplifying the equation, it means that the equation is an identity and has infinitely many solutions.

Answer:

infinitely many