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 - hand side

First, we expand the expression \(3(4x + 1)-2\) using the distributive property \(a(b + c)=ab+ac\). So, \(3(4x + 1)=3\times4x+3\times1 = 12x+3\). Then \(3(4x + 1)-2=12x + 3-2=12x+1\).

Step2: Analyze the equation

The original equation is \(12x + 1=3(4x + 1)-2\). After expanding the right - hand side, we get \(12x+1 = 12x + 1\).
Subtract \(12x\) from both sides of the equation: \((12x + 1)-12x=(12x + 1)-12x\).
Simplifying both sides, we have \(1 = 1\). This is a true statement for all values of \(x\).

Answer:

infinitely many