1. Problem 15: Solve the equation $a + 80 = 8a$. Step 1: Subtract $a$ from both sides to isolate terms with $a$ on one side: $$a + 80 = 8a \implies 80 = 8a - a$$ Step 2: Simplify the right side: $$80 = 7a$$ Step 3: Divide both sides by 7 to solve for $a$: $$a = \frac{80}{7}$$ Final answer for problem 15: $a = \frac{80}{7}$. 2. Problem 16: Simplify the expression $$\frac{-16^2 + \sqrt{16^2 - 4(3)(6)}}{2}$$. Step 1: Calculate $-16^2$ carefully. Note that $-16^2 = -(16^2) = -256$. Step 2: Calculate the discriminant inside the square root: $$16^2 - 4(3)(6) = 256 - 72 = 184$$ Step 3: Substitute back: $$\frac{-256 + \sqrt{184}}{2}$$ Step 4: Simplify $\sqrt{184}$. Since $184 = 4 \times 46$, $$\sqrt{184} = \sqrt{4 \times 46} = 2\sqrt{46}$$ Step 5: Write the expression as: $$\frac{-256 \pm 2\sqrt{46}}{2}$$ Step 6: Divide numerator terms by 2: $$-128 \pm \sqrt{46}$$ Final answer for problem 16: $$-128 \pm \sqrt{46}$$. 3. Problem 17: Solve the inequality $$\frac{5}{x+2} > \frac{5}{x} + \frac{2}{3x}$$. Step 1: Find a common denominator for the right side terms: $$\frac{5}{x} + \frac{2}{3x} = \frac{15}{3x} + \frac{2}{3x} = \frac{17}{3x}$$ Step 2: Rewrite the inequality: $$\frac{5}{x+2} > \frac{17}{3x}$$ Step 3: Cross-multiply, noting the domain restrictions $x \neq 0$, $x \neq -2$: $$5 \cdot 3x > 17(x+2)$$ Step 4: Simplify both sides: $$15x > 17x + 34$$ Step 5: Subtract $17x$ from both sides: $$15x - 17x > 34 \implies -2x > 34$$ Step 6: Divide both sides by $-2$, reversing the inequality sign: $$x < -17$$ Step 7: Consider domain restrictions: $x \neq 0$, $x \neq -2$. Since $x < -17$ does not include these values, the solution is valid. Final answer for problem 17: $$x < -17$$.