Solve the inequality -6(x + 3) $\leq$ 4(x - 2)

Analysis

To solve the inequality $-6(x + 3) \leq 4(x - 2)$, we need to expand the brackets, collect like terms, and isolate the variable $x$. A crucial rule to remember is that when you divide or multiply an inequality by a negative number, the inequality sign must be reversed.

Concept

1. Expansion: Use the distributive property to remove the brackets.
2. Grouping: Move all terms containing $x$ to one side and constant terms to the other.
3. Division: Divide by the coefficient of $x$. If the coefficient is negative, flip the inequality sign ($\leq$ becomes $\geq$).

Step-by-Step Solution

Step 1: Expand the brackets $-6 \times x - 6 \times 3 \leq 4 \times x - 4 \times 2$ $-6x - 18 \leq 4x - 8$

Step 2: Collect like terms Move $4x$ to the left side and $-18$ to the right side: $-6x - 4x \leq -8 + 18$

Step 3: Simplify both sides $-10x \leq 10$

Step 4: Solve for $x$ Divide both sides by $-10$. Because we are dividing by a negative number, the inequality sign $\leq$ changes to $\geq$: $x \geq \frac{10}{-10}$ $x \geq -1$

Conclusion

Comparing our result ($x \geq -1$) with the given options, it matches option B.

Exam Tip: Always double-check your sign when dividing by a negative number! Many students lose marks here by forgetting to flip the inequality sign. If you are ever unsure, pick a number that satisfies your final answer (e.g., $x = 0$) and plug it back into the original equation to see if it holds true.