Let’s start by breaking down each step with detailed reasons for the corresponding statement to solve the given equation [tex]\(4x + 1 = 6x - 2\)[/tex] and prove that [tex]\(x = \frac{3}{2}\)[/tex].
### Detailed Solution:
[tex]\[
\begin{tabular}{|l|l|}
\hline \multicolumn{1}{|c|}{Statement} & \multicolumn{1}{|c|}{Reason} \\
\hline 1. \(4x + 1 = 6x - 2\) & Given equation \\
\hline 2. \(4x + 1 + 2 = 6x\) & Addition property of equality (adding 2 to both sides) \\
\hline 3. \(4x + 3 = 6x\) & Simplifying both sides (combining like terms) \\
\hline 4. \(4x - 4x + 3 = 6x - 4x\) & Subtraction property of equality (subtracting \(4x\) from both sides) \\
\hline 5. \(3 = 2x\) & Simplifying both sides (combining like terms) \\
\hline 6. \(\frac{3}{2} = x\) & Division property of equality (dividing both sides by 2) \\
\hline 7. \(x = \frac{3}{2}\) & Symmetry property of equality (rewriting equation) \\
\hline
\end{tabular}
\][/tex]
### Proof Outline:
1. Given Equation:
[tex]\[ 4x + 1 = 6x - 2 \][/tex]
2. Isolate Terms with [tex]\(x\)[/tex] one side:
Add 2 to both sides of the equation to shift the constant term from the right-hand side to the left-hand side.
[tex]\[ 4x + 1 + 2 = 6x - 2 + 2 \][/tex]
Simplify both sides:
[tex]\[ 4x + 3 = 6x \][/tex]
3. Isolate Variable [tex]\(x\)[/tex]:
Subtract [tex]\(4x\)[/tex] from both sides of the equation to consolidate all [tex]\(x\)[/tex]-terms on one side.
[tex]\[ 4x + 3 - 4x = 6x - 4x \][/tex]
Simplify both sides:
[tex]\[ 3 = 2x \][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by 2 to solve for [tex]\(x\)[/tex].
[tex]\[ \frac{3}{2} = x \][/tex]
5. Rewrite Solution Symmetrically:
To present the final answer in a traditional format, rewrite:
[tex]\[ x = \frac{3}{2} \][/tex]
This completes the proof. The solve for [tex]\(x\)[/tex] yields [tex]\(x = 1.5\)[/tex], or equivalently [tex]\(x = \frac{3}{2}\)[/tex].
