(a)

\begin{align} 13x & < - 5x + 2 \\ 13x + 5x & < 2 \\ 18x & < 2 \\ x & < {2 \over 18} \\ x & < {1 \over 9} \end{align}

(b)

\begin{align} x - 6 & > 4x - 9 \\ -6 + 9 & > 4x - x \\ 3 & > 3x \\ {3 \over 3} & > x \\ 1 & > x \end{align}

(c)

\begin{align} 7(x - 3) & \le 2x + 1 \\ 7x - 21 & \le 2x + 1 \\ 7x - 2x & \le 1 + 21 \\ 5x & \le 22 \\ x & \le {22 \over 5} \end{align}

(d)

\begin{align} 8x + 3 & \ge 4x + 7 \\ 8x - 4x & \ge 7 - 3 \\ 4x & \ge 4 \\ x & \ge {4 \over 4} \\ x & \ge 1 \end{align}

(a)

\begin{align} 5 - a & \le a - 6 &&& a - 6 & \le 10 - 3a \\ - a - a & \le -6 - 5 &&& a + 3a & \le 10 + 6 \\ -2a & \le -11 &&& 4a & \le 16 \\ a & \ge {-11 \over -2} &&& a & \le {16 \over 4} \\ a & \ge 5.5 &&& a & \le 4 \end{align}

$$ \text{No solutions} $$

(b)

\begin{align} 4 - b & < 2b - 1 &&& 2b - 1 & < 7 + b \\ 4 + 1 & < 2b + b &&& 2b - b & < 7 + 1 \\ 5 & < 3b &&& b & < 8 \\ {5 \over 3} & < b &&& b & < 8 \end{align}
$$ {5 \over 3} < b < 8 $$

(c)

\begin{align} 4c - 1 & < {1 \over 2} &&& {1 \over 2} & \le 3c + 2 \\ 4c & < {1 \over 2} + 1 &&& {1 \over 2} - 2 & \le 3c \\ 4c & < {3 \over 2} &&& -{3 \over 2} & \le 3c \\ c & < {1 \over 4} \left(3 \over 2\right) &&& {1 \over 3}\left(-{3 \over 2}\right) & \le c \\ c & < {3 \over 8} &&& -{1 \over 2} & \le c \end{align}
$$ -{1 \over 2} \le c < {3 \over 8} $$

(d)

\begin{align} 2d + 1 & \ge d &&& d & > 3d - 20 \\ 2d - d & \ge -1 &&& 20 & > 3d - d \\ d & \ge - 1 &&& 20 & > 2d \\ & &&& {20 \over 2} & > d \\ & &&& 10 & > d \end{align}
$$ -1 \le d < 10 $$

(a)

\begin{align} 5x & > 69 - 2x &&& 27 - 2x & \ge 4 \\ 5x + 2x & > 69 &&& 27 - 4 & \ge 2x \\ 7x & > 69 &&& 23 & \ge 2x \\ x & > {69 \over 7} &&& {23 \over 2} & \ge x \\ x & > 9{6 \over 7} &&& 11{1 \over 2} & \ge x \end{align}
\begin{align} 9{6 \over 7} & \phantom{.} < x < 11{1 \over 2} \\ \\ \text{Integer values} & \text{ of } x \text{ are 10 and 11} \end{align}

(b)

\begin{align} -10 & \le x < - 4 &&& 2 - 5x & < 35 \\ & &&& -5x & < 35 - 2 \\ & &&& -5x & < 33 \\ & &&& x & > {33 \over -5} \\ & &&& x & > -6{3 \over 5} \end{align}
\begin{align} -6{3 \over 5} & \phantom{.} < x < -4 \\ \\ \text{Integer values} & \text{ of } x \text{ are -6 and -5} \end{align}

(a)

\begin{align} 3x - 5 & < 26 &&& 26 & \le 4x - 6 \\ 3x & < 26 + 5 &&& 26 + 6 & \le 4x \\ 3x & < 31 &&& 32 & \le 4x \\ x & < {31 \over 3} &&& {32 \over 4} & \le x \\ x & < 10{1 \over 3} &&& 8 & \le x \end{align}
\begin{align} 8 & \phantom{.} \le x < 10{1 \over 3} \\ \\ \text{Integer values} & \text{ of } x \text{ are 8, 9 and 10} \end{align}

(b)

\begin{align} 3x + 2 & < 19 &&& 19 & < 5x - 4 \\ 3x & < 19 - 2 &&& 19 + 4 & < 5x \\ 3x & < 17 &&& 23 & < 5x \\ x & < {17 \over 3} &&& {23 \over 5} & < x \\ x & < 5{2 \over 3} &&& 4{3 \over 5} & < x \end{align}
\begin{align} 4{3 \over 5} & \phantom{.} \le x < 5{2 \over 3} \\ \\ \text{Integer values} & \text{ of } x \text{ is 5} \end{align}

(c)

\begin{align} 3 & < 3x - 9 &&& 3x - 9 & < 4 \\ 3 + 9 & < 3x &&& 3x & < 4 + 9 \\ 12 & < 3x &&& 3x & < 13 \\ {12 \over 3} & < x &&& x & < {13 \over 3} \\ 4 & < x &&& x & < 4{1 \over 3} \end{align}
\begin{align} 4 & \phantom{.} < x < 4{1 \over 3} \\ \\ \text{No integers} & \text{ satisfy the inequality} \end{align}

(d)

\begin{align} -10 & < 7 - 2x &&& 7 - 2x & \le -1 \\ 2x & < 7 + 10 &&& 7 + 1 & \le 2x \\ 2x & < 17 &&& 8 & \le 2x \\ x & < {17 \over 2} &&& {8 \over 2} & \le x \\ x & < 8{1 \over 2} &&& 4 & \le x \end{align}
\begin{align} 4 & \phantom{.} \le x < 8{1 \over 2} \\ \\ \text{Integer values} & \text{ of } x \text{ are 4, 5, 6, 7 and 8} \end{align}

\begin{align} \text{Let } x \text{ represent } & \text{the prime number} \end{align}
\begin{align} 3(x + 5) - 7 & \le 55 &&& 4(x - 2) & > 39 \\ 3x + 15 - 7 & \le 55 &&& 4x - 8 & > 39 \\ 3x + 8 & \le 55 &&& 4x & > 39 + 8 \\ 3x & \le 55 - 8 &&& 4x & > 47 \\ 3x & \le 47 &&& x & > {47 \over 4} \\ x & \le {47 \over 3} &&& x & > 11{3 \over 4} \\ x & \le 15{2 \over 3} \end{align}
\begin{align} 11{3 \over 4} & \phantom{.} < x \le 15{2 \over 3} \\ \\ \text{The prime } & \text{number is } 13 \end{align}

To solve this question, we assume that each of the customers spend the minimum $20 (worst case scenario) in order to receive the goodie bag

\begin{align} 45 \text{ cents} & = \$ 0.45 \\ \\ 20 \text{ cents} & = \$ 0.20 \\ \\ \text{Cost of 1 goodie bag} & = 2(0.45) + 3(0.20) \\ & = \$ 1.50 \end{align}
\begin{align} \text{Total cost of goodie bags} & \le \$200 &&& \text{Total sales} & \ge \$2000 \\ (1.50)(x) & \le 200 &&& (20)(x) & \ge 2000 \\ 1.5x & \le 200 &&& 20x & \ge 2000 \\ x & \le {200 \over 1.5} &&& x & \ge {2000 \over 20} \\ x & \le 133{1 \over 3} &&& x & \ge 100 \end{align}
\begin{align} 100 & \phantom{.} \le x \le 133{1 \over 3} \\ \\ \text{Possible } & \text{value of } x \text{ is 100} \end{align}