Evaluate 3(x + 2) > 6(x + 3)

Analysis

We are asked to solve a linear inequality. The goal is to isolate the variable $x$ on one side of the inequality sign to determine its range of values.

Concept

When solving inequalities, the rules are very similar to solving equations, with one critical exception: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.

Step-by-Step Solution

Step 1: Expand the brackets Distribute the numbers outside the parentheses: $3(x) + 3(2) > 6(x) + 6(3)$ $3x + 6 > 6x + 18$

Step 2: Group the like terms Move the $x$ terms to one side and the constant numbers to the other. Let's move $3x$ to the right and $18$ to the left: $6 - 18 > 6x - 3x$

Step 3: Simplify both sides $-12 > 3x$

Step 4: Solve for $x$ Divide both sides by $3$: $\frac{-12}{3} > \frac{3x}{3}$ $-4 > x$

This can also be written as: $x < -4$

Conclusion

By simplifying the inequality, we found that $x$ must be less than $-4$. This matches option C.

Exam Tip

When you have an inequality like $a > b$, it is exactly the same as $b < a$. If you ever feel confused about which way the sign should face, read it out loud: "$-4$ is greater than $x$" is the same as saying "$x$ is less than $-4$." Always double-check your sign direction after dividing by a negative number!