Solve for x: 3(x – 1) ≤ 2 (x – 3)

Analysis

This is a linear inequality problem. The goal is to isolate the variable $x$ on one side of the inequality sign, similar to how you would solve a linear equation, while being mindful of the rules regarding inequality signs.

Concept

To solve linear inequalities:

1. Expand the brackets: Use the distributive property to multiply the terms inside the parentheses.
2. Collect like terms: Move all terms containing $x$ to one side and constant numbers to the other.
3. Simplify: Combine the terms.
4. Solve for $x$: If you divide or multiply by a negative number, remember to flip the inequality sign. In this case, we will see if that is necessary.

Step-by-Step Solution

Step 1: Expand the brackets Given: $3(x - 1) \leq 2(x - 3)$

Multiply $3$ by $(x - 1)$ and $2$ by $(x - 3)$: $3x - 3 \leq 2x - 6$

Step 2: Collect like terms Subtract $2x$ from both sides to bring the $x$ terms to the left: $3x - 2x - 3 \leq -6$ $x - 3 \leq -6$

Step 3: Isolate $x$ Add $3$ to both sides of the inequality: $x \leq -6 + 3$ $x \leq -3$

Conclusion

After simplifying the inequality, we arrive at $x \leq -3$. This matches option A.

Exam Tip

When solving inequalities, always double-check your signs. A common mistake students make is forgetting to flip the inequality sign when dividing by a negative number (e.g., if you had $-x \leq 3$, it would become $x \geq -3$). In this specific problem, we did not need to flip the sign because the coefficient of $x$ remained positive.