Find all real number x which satisfy the inequality $\frac{1}{3}$ (x + 1) - 1 > $\frac{1}{5}$(x + 4)

To solve the inequality $\frac{1}{3}(x + 1) - 1 > \frac{1}{5}(x + 4)$, we will follow these steps:

1. Analyze the inequality

We have a linear inequality involving fractions. To make the calculation easier, we should eliminate the fractions by multiplying the entire inequality by the Least Common Multiple (LCM) of the denominators (3 and 5).

The LCM of 3 and 5 is 15.

2. Step-by-step solution

Step 1: Multiply every term by 15 to clear the fractions: $15 \times [\frac{1}{3}(x + 1)] - 15 \times [1] > 15 \times [\frac{1}{5}(x + 4)]$

Step 2: Simplify the terms: $5(x + 1) - 15 > 3(x + 4)$

Step 3: Expand the brackets: $5x + 5 - 15 > 3x + 12$

Step 4: Combine like terms on the left side: $5x - 10 > 3x + 12$

Step 5: Group the $x$ terms on one side and the constants on the other: Subtract $3x$ from both sides: $5x - 3x - 10 > 12$ $2x - 10 > 12$

Add 10 to both sides: $2x > 12 + 10$ $2x > 22$

Step 6: Solve for $x$: Divide both sides by 2: $x > \frac{22}{2}$ $x > 11$

3. Conclusion

Comparing our result ($x > 11$) with the given options, we find that it matches option D.

Exam Tip

When dealing with inequalities involving fractions, always multiply by the LCM of the denominators first. This turns the problem into a simple linear inequality and helps you avoid errors with fraction arithmetic. Also, remember that if you multiply or divide an inequality by a negative number, you must reverse the inequality sign (e.g., $>$ becomes $<$). In this problem, we only multiplied by a positive number (15), so the sign remained the same.