Solve for x in $\frac{4x - 6}{3} \leq \frac{3 + 2x}{2}$

Analysis

To solve this linear inequality, we need to eliminate the fractions by finding a common denominator and then isolate the variable $x$.

Concept

When dealing with inequalities involving fractions, the most efficient method is to multiply the entire inequality by the Least Common Multiple (LCM) of the denominators. In this case, the denominators are 3 and 2, so the LCM is 6.

Important Rule: When multiplying or dividing an inequality by a negative number, you must reverse the inequality sign. Since we are multiplying by a positive 6, the sign remains the same.

Step-by-Step Solution

Step 1: Multiply both sides by 6 to clear the fractions.

Step 2: Simplify the expression.

Step 3: Expand the brackets.

Step 4: Group the $x$ terms on one side and the constants on the other. Subtract $6x$ from both sides:

Add 12 to both sides:

Step 5: Solve for $x$. Divide both sides by 2:

Or, in mixed fraction form:

Conclusion

Comparing our result ($x \leq \frac{21}{2}$) with the provided options, it matches option B.

Exam Tip

Always check your answer by picking a number that satisfies your result. For example, if $x = 0$ (which is $\leq 10.5$), the original inequality becomes $\frac{-6}{3} \leq \frac{3}{2}$, which is $-2 \leq 1.5$. Since this is true, your solution is likely correct!