Let's solve the inequality step by step:

The given inequality is: $5x - 3(x + 6) \geq 8 - (x - 7)$

Step 1: Expand both sides of the inequality

On the left side, distribute the $-3$ through $(x + 6)$: $5x - 3(x + 6) = 5x - 3x - 18 = 2x - 18$

On the right side, distribute the negative sign through $(x - 7)$: $8 - (x - 7) = 8 - x + 7 = 15 - x$

Now the inequality becomes: $2x - 18 \geq 15 - x$

Step 2: Move all terms involving $x$ to one side

Add $x$ to both sides to eliminate the $-x$ on the right: $2x + x - 18 \geq 15$ $3x - 18 \geq 15$

Step 3: Solve for $x$

Add 18 to both sides: $3x \geq 33$

Divide by 3: $x \geq 11$

Conclusion:

The solution is $x \geq 11$, which corresponds to the last option: $\boxed{x \geq 11}$

Would you like further details or have any questions?

Related questions:

1. How do you solve inequalities involving absolute values?
2. How can we graph the solution to this inequality on a number line?
3. What are the differences between solving an inequality and solving an equation?
4. How does the inequality change when multiplying or dividing by a negative number?
5. Can inequalities have more than one solution range?

Tip:

Always remember, when multiplying or dividing both sides of an inequality by a negative number, the inequality symbol must be flipped.