If the linear equation 5x + 6(3 − 2x) = 4 holds for a real number x, what is the value of x that satisfies this equation?

Introduction / Context:
This is a straightforward linear equation involving a bracket. The task is to expand the bracket, combine like terms, and solve for the variable x. Such questions test basic algebra skills, especially the correct application of the distributive property and manipulation of equations.

Given Data / Assumptions:

• The equation is 5x + 6(3 − 2x) = 4.
• x is a real number.
• We must find the value of x that makes this equality true.
• Standard algebraic operations such as expansion and simplification are allowed.

Concept / Approach:
The main steps are: expand the bracket 6(3 − 2x), then collect all terms involving x on one side of the equation and constants on the other side. This will give a simple equation of the form kx = constant. Solving this equation yields the value of x. Care must be taken with the signs when distributing the 6 across the bracket.

Step-by-Step Solution:
Start with 5x + 6(3 − 2x) = 4.Expand the bracket: 6 * 3 = 18 and 6 * (−2x) = −12x.So the equation becomes 5x + 18 − 12x = 4.Combine like terms for x: 5x − 12x = −7x, giving −7x + 18 = 4.Subtract 18 from both sides: −7x = 4 − 18 = −14.Divide both sides by −7: x = (−14) / (−7) = 2.

Verification / Alternative check:
Substitute x = 2 back into the original equation. Compute the bracket first: 3 − 2x = 3 − 4 = −1. Then 6(3 − 2x) = 6(−1) = −6. Now compute 5x + 6(3 − 2x) = 5 * 2 + (−6) = 10 − 6 = 4, which matches the right-hand side. Therefore x = 2 is indeed the correct solution.

Why Other Options Are Wrong:

• For x = 1, the left side becomes 5 * 1 + 6(3 − 2) = 5 + 6 = 11, not 4.
• For x = 3, LHS = 15 + 6(3 − 6) = 15 − 18 = −3, not 4.
• For x = 4, LHS = 20 + 6(3 − 8) = 20 − 30 = −10.
• For x = 0, LHS = 0 + 18 = 18, again not equal to 4.

Common Pitfalls:

• Incorrectly expanding 6(3 − 2x), sometimes writing 18 + 12x instead of 18 − 12x.
• Not combining the x terms correctly when simplifying 5x − 12x.
• Dropping the negative sign when dividing by −7, which would give x = −2 instead of 2.

Final Answer:
2