Solve the linear equation 5x − 6 = 3x − 8 and choose the correct value of x.

Introduction / Context:
This is a simple linear equation in one variable. Solving such equations is a fundamental algebraic skill that involves isolating the variable on one side of the equation using inverse operations while maintaining equality.

Given Data / Assumptions:

• Equation: 5x − 6 = 3x − 8.
• x is a real number.
• We can add, subtract, multiply or divide both sides by the same nonzero quantity to isolate x.

Concept / Approach:
The standard approach is to move all x terms to one side and all constant terms to the other side. This is done by adding or subtracting the same expressions from both sides. After that, we solve the resulting simple equation for x by dividing by the coefficient of x.

Step-by-Step Solution:
Start with 5x − 6 = 3x − 8.Subtract 3x from both sides to bring x terms together: 5x − 3x − 6 = −8.This simplifies to 2x − 6 = −8.Add 6 to both sides to move the constant: 2x = −8 + 6 = −2.Divide both sides by 2: x = −2 ÷ 2 = −1.

Verification / Alternative check:
Substitute x = −1 back into the original equation.Left-hand side: 5(−1) − 6 = −5 − 6 = −11.Right-hand side: 3(−1) − 8 = −3 − 8 = −11.Since both sides are equal, x = −1 is indeed a correct solution.

Why Other Options Are Wrong:
If x = 2, left-hand side is 10 − 6 = 4 while right-hand side is 6 − 8 = −2, which are not equal.If x = −2, left-hand side is −10 − 6 = −16 while right-hand side is −6 − 8 = −14, which again do not match.If x = 1, left-hand side is 5 − 6 = −1 while right-hand side is 3 − 8 = −5, so the equality fails.

Common Pitfalls:
A common mistake is to move terms across the equals sign without changing their sign, which leads to an incorrect equation.Another pitfall is to perform incorrect arithmetic when combining the constant terms or when dividing by the coefficient of x.

Final Answer:
The solution of the equation 5x − 6 = 3x − 8 is x = −1.