Let's solve the inequality $1 + 4\left(-\frac{1}{2}x - \frac{1}{3}\right) < -5 - x$.

Step 1: Distribute the 4

First, distribute the 4 across the terms inside the parentheses: $1 + 4\left(-\frac{1}{2}x\right) + 4\left(-\frac{1}{3}\right) < -5 - x$ This simplifies to: $1 - 2x - \frac{4}{3} < -5 - x$

Step 2: Combine like terms on the left-hand side

Next, combine the constants on the left-hand side: $1 - \frac{4}{3} - 2x < -5 - x$ $\frac{3}{3} - \frac{4}{3} - 2x < -5 - x$ $-\frac{1}{3} - 2x < -5 - x$

Step 3: Add $2x$ to both sides to simplify

Add $2x$ to both sides to isolate the variable $x$: $-\frac{1}{3} < -5 + x$

Step 4: Add 5 to both sides

Finally, add 5 to both sides to further isolate $x$: $-\frac{1}{3} + 5 < x$

Step 5: Simplify

$\frac{15}{3} - \frac{1}{3} < x$ $\frac{14}{3} < x$ Or equivalently, $x > \frac{14}{3}$

Final Answer:

The solution to the inequality is: $x > \frac{14}{3}$

Do you want details or have any questions?
Here are 5 related questions:

1. How do you solve inequalities with fractions?
2. What happens when you multiply or divide both sides of an inequality by a negative number?
3. Can you explain how to combine like terms in inequalities?
4. How do you graph the solution $x > \frac{14}{3}$ on a number line?
5. What is the difference between solving equations and solving inequalities?

Tip: Always remember that when you multiply or divide an inequality by a negative number, you must flip the inequality sign.