1. Problem: Simplify $\frac{6-7i}{1-2i}$. 2. Rule: Multiply numerator and denominator by the complex conjugate of the denominator to get a real denominator. 3. Work: $\frac{6-7i}{1-2i}=\frac{(6-7i)(1+2i)}{(1-2i)(1+2i)}$. 4. Work: Numerator expansion $=(6+12i-7i-14i^2)=20+5i$. 5. Work: Denominator $=(1-2i)(1+2i)=1+4=5$. 6. Result: $\frac{20+5i}{5}=4+i$. 7. Problem: Given one root $w_1=-1+2i$ of $f(x)=x^2+2x+5$, find the other root $w_2$ and state the relationship between $w_1$ and $w_2$. 8. Rule: For $x^2+2x+5=0$ the sum of roots is $-2$ and the product is $5$ by Vieta's formulas. 9. Work: $w_2=\text{sum}-w_1=-2-(-1+2i)=-1-2i$. 10. Relationship: $w_2$ is the complex conjugate of $w_1$, so $w_2=\overline{w_1}=-1-2i$, and $w_1w_2=5$. 11. Problem: Determine the modulus of $w_1=-1+2i$. 12. Rule: Modulus $|a+bi|=\sqrt{a^2+b^2}$. 13. Work: $|w_1|=\sqrt{(-1)^2+2^2}=\sqrt{1+4}=\sqrt{5}$. 14. Result: $|w_1|=\sqrt{5}$. 15. Problem: Solve the inequality $|3x+2|\le 5$. 16. Rule: $|A|\le B$ with $B\ge0$ is equivalent to $-B\le A\le B$. 17. Work: $-5\le 3x+2\le 5$. 18. Work: Subtract 2: $-7\le 3x\le 3$. 19. Work: Divide by 3: $-\frac{7}{3}\le x\le 1$. 20. Result: $x\in [-\frac{7}{3},1]$. 21. Problem: Find the coefficient of the $x$ term in the expansion of $\left(5-\frac{x}{3}\right)^3$. 22. Rule: Use the binomial expansion $ (a-b)^3=a^3-3a^2b+3ab^2-b^3$. 23. Work: With $a=5$ and $b=\frac{x}{3}$ the linear-in-$x$ term is $-3a^2b=-3\cdot25\cdot\frac{x}{3}$. 24. Work: That equals $-75\cdot\frac{x}{3}=-25x$. 25. Result: The coefficient of $x$ is $-25$. 26. Problem (d)(i): From purchases state two linear equations for the other purchases. 27. Given: Second purchase 3 pencils, 1 pen, 1 eraser for 33 gives $3x+y+z=33$. 28. Given: Third purchase 1 pencil and 4 pens for 45 gives $x+4y+0z=45$. 29. Problem (d)(ii): Construct the augmented matrix $[A|b]$ for the three equations $4x+3y+3z=74$, $3x+y+z=33$, $x+4y+0z=45$. 30. Work: Coefficient matrix $A=\begin{bmatrix}4 & 3 & 3\\3 & 1 & 1\\1 & 4 & 0\end{bmatrix}$ and column $b=\begin{bmatrix}74\\33\\45\end{bmatrix}$. 31. Result: Augmented matrix $$[A|b]=\begin{bmatrix}4 & 3 & 3 & | & 74\\3 & 1 & 1 & | & 33\\1 & 4 & 0 & | & 45\end{bmatrix}$$. 32. Problem 2.(a)(i): Solve $\dfrac{9^{2x+1}}{3^x}=1$. 33. Rule: Write $9=3^2$ so $9^{2x+1}=3^{4x+2}$ and use $3^{\text{exponent}}=1\iff$ exponent $=0$. 34. Work: Equation becomes $3^{4x+2-x}=3^{3x+2}=1$. 35. Work: So $3x+2=0$ and $x=-\frac{2}{3}$. 36. Result: $x=-\frac{2}{3}$. 37. Problem 2.(a)(ii): Solve $\log(3x+1)=2$. 38. Rule: Assume base 10, so $3x+1=10^2=100$. 39. Work: $3x=99$. 40. Result: $x=33$. 41. Problem 2.(a)(iii): Prove $f\left(\dfrac{2x}{1+x^2}\right)=2f(x)$ for $f(x)=\ln\left(\dfrac{1+x}{1-x}\right)$ with $-1