Question

which equation has no solution?
4(x + 3) + 2x = 6(x + 2)
5 + 2(3 + 2x) = x + 3(x + 1)
5(x + 3) + x = 4(x + 3) + 3
4 + 6(2 + x) = 2(3x + 8)

Explanation:

Step1: Analyze first equation \(4(x + 3)+2x = 6(x + 2)\)

Expand both sides: \(4x+12 + 2x=6x + 12\)
Combine like terms: \(6x+12=6x + 12\)
Subtract \(6x\) and \(12\) from both sides: \(0 = 0\), infinite solutions.

Step2: Analyze second equation \(5 + 2(3 + 2x)=x + 3(x + 1)\)

Expand both sides: \(5+6 + 4x=x + 3x+3\)
Combine like terms: \(11 + 4x=4x + 3\)
Subtract \(4x\) from both sides: \(11=3\), which is false, no solution.

Step3: Analyze third equation \(5(x + 3)+x = 4(x + 3)+3\)

Expand both sides: \(5x+15 + x=4x+12 + 3\)
Combine like terms: \(6x+15=4x + 15\)
Subtract \(4x\) and \(15\): \(2x=0\), \(x = 0\), one solution.

Step4: Analyze fourth equation \(4 + 6(2 + x)=2(3x + 8)\)

Expand both sides: \(4+12 + 6x=6x + 16\)
Combine like terms: \(16 + 6x=6x + 16\)
Subtract \(6x\) and \(16\): \(0 = 0\), infinite solutions.

Answer:

\(5 + 2(3 + 2x)=x + 3(x + 1)\) (the second equation)