1. **State the problem:** Solve the inequality $$5x - 3 \leq 3x + 11$$ and represent the solution on a number line and as a solution set. 2. **Write the inequality:** $$5x - 3 \leq 3x + 11$$ 3. **Isolate variable terms on one side:** Subtract $$3x$$ from both sides: $$5x - 3x - 3 \leq 3x - 3x + 11$$ which simplifies to $$2x - 3 \leq 11$$ 4. **Isolate the constant term:** Add $$3$$ to both sides: $$2x - 3 + 3 \leq 11 + 3$$ which simplifies to $$2x \leq 14$$ 5. **Solve for $$x$$:** Divide both sides by $$2$$: $$\frac{\cancel{2}x}{\cancel{2}} \leq \frac{14}{2}$$ which simplifies to $$x \leq 7$$ 6. **Interpretation:** The solution set is all real numbers $$x$$ such that $$x \leq 7$$. 7. **Solution set notation:** $$\{x \mid x \leq 7\}$$ 8. **Number line representation:** Shade all points to the left of and including $$7$$. **Final answer:** $$x \leq 7$$