1. **State the problem:** Aaliyah runs and bikes for a total of 2 hours. She runs 4 miles and bikes 8 miles. Her bike speed is 4 mph faster than her running speed. We need to find her running speed in miles per hour. 2. **Define variables:** Let $r$ be her running speed in mph. Then her biking speed is $r + 4$ mph. 3. **Use the formula for time:** Time = Distance / Speed. 4. **Write the total time equation:** Running time + Biking time = 2 hours. $$\frac{4}{r} + \frac{8}{r+4} = 2$$ 5. **Solve the equation:** Multiply both sides by $r(r+4)$ to clear denominators: $$4(r+4) + 8r = 2r(r+4)$$ Simplify: $$4r + 16 + 8r = 2r^2 + 8r$$ Combine like terms: $$12r + 16 = 2r^2 + 8r$$ Bring all terms to one side: $$0 = 2r^2 + 8r - 12r - 16$$ $$0 = 2r^2 - 4r - 16$$ Divide entire equation by 2: $$0 = r^2 - 2r - 8$$ 6. **Factor or use quadratic formula:** $$r = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-8)}}{2} = \frac{2 \pm \sqrt{4 + 32}}{2} = \frac{2 \pm \sqrt{36}}{2}$$ $$r = \frac{2 \pm 6}{2}$$ 7. **Find possible values:** $$r = \frac{2 + 6}{2} = 4$$ or $$r = \frac{2 - 6}{2} = -2$$ Since speed cannot be negative, $r = 4$ mph. **Final answer:** Aaliyah runs at 4 miles per hour.