(1) \( x+1=b(b-x) \) \( \begin{array}{ll}\text { (3) } 4 a x-4 b x=a^{2}-2 a b+b^{2} & \text { (4) } \frac{x^{2}}{c^{2}}=b^{2} \\ \text { (5) } \frac{x}{a}-3=5-\frac{x}{b} & \text { (6) } \frac{x+a}{x-b}=\frac{c}{3} \\ \text { (7) } \frac{a}{x-1}=\frac{b}{x+2} & \text { (8) } \frac{2 x}{x+a}=\frac{b}{x-1}+2 \\ \text { (g) } \text { Los op vir } x \text { en stel jou antwoord op 'n getallelyn voor: } \\ \text { (1) } 2 x+1 \geq 13 & \text { (2) } 2-2 x \leq 3 \\ \text { (3) } 3(x+7)>6 & \text { (4) } 10-2 x<22+2 x \\ \text { (5) } \frac{1-x}{2} \leq 3-x & \text { (6) } \frac{-5 x}{3}-1 \geq x+7 \\ \text { Los op vir } x \text { en stel jou antwoord op 'n getallelyn voor: } \\ \text { (1) }-4 \leq-2 x<6 & \text { (2) }-8 \leq 4 x-12<0 \\ \text { (3) }-7 \leq 3 x-1<5 & \text { (4) }-2 \leq 4-2 x \leq 8 \\ \text { (5) }-4<\frac{3 x-5}{2}<8 & \text { (6) }-5<\frac{1-2 x}{3} \leq 3\end{array} \)

Solve the inequality by following steps: - step0: Solve for \(x\): \(2 x+1 \geq 13;2-2 x \leq 3;3(x+7)>6;10-2 x<22+2 x;\frac{1-x}{2} \leq 3-x;\frac{-5 x}{3}-1 \geq x+7;\) - step1: Separate into two inequalities: \(\left\{ \begin{array}{l}2x+1\geq 13\\2-2x\leq 3\\3\left(x+7\right)>6\\10-2x<22+2x\\\frac{1-x}{2}\leq 3-x\\\frac{-5x}{3}-1\geq x+7\end{array}\right.\) - step2: Solve the inequality: \(\left\{ \begin{array}{l}x\geq 6\\x\geq -\frac{1}{2}\\x>-5\\x>-3\\x\leq 5\\x\leq -3\end{array}\right.\) - step3: Find the intersection: \(x \in \varnothing \) Solve the system of equations \( -4 \leq-2 x<6;-8 \leq 4 x-12<0;-7 \leq 3 x-1<5;-2 \leq 4-2 x \leq 8;-4<\frac{3 x-5}{2}<8;-5<\frac{1-2 x}{3} \leq 3; \). Solve the inequality by following steps: - step0: Solve for \(x\): \(-4 \leq-2 x<6;-8 \leq 4 x-12<0;-7 \leq 3 x-1<5;-2 \leq 4-2 x \leq 8;-4<\frac{3 x-5}{2}<8;-5<\frac{1-2 x}{3} \leq 3;\) - step1: Separate into two inequalities: \(\left\{ \begin{array}{l}-4\leq -2x\\-2x<6\\-8\leq 4x-12\\4x-12<0\\-7\leq 3x-1\\3x-1<5\\-2\leq 4-2x\\4-2x\leq 8\\-4<\frac{3x-5}{2}\\\frac{3x-5}{2}<8\\-5<\frac{1-2x}{3}\\\frac{1-2x}{3}\leq 3\end{array}\right.\) - step2: Solve the inequality: \(\left\{ \begin{array}{l}x\leq 2\\x>-3\\x\geq 1\\x<3\\x\geq -2\\x<2\\x\leq 3\\x\geq -2\\x>-1\\x<7\\x<8\\x\geq -4\end{array}\right.\) - step3: Simplify: \(\left\{ \begin{array}{l}x\leq 2\\x>-3\\x\geq 1\\x<3\\x\geq -2\\x<2\\x\leq 3\\x>-1\\x<7\\x<8\\x\geq -4\end{array}\right.\) - step4: Find the intersection: \(1\leq x<2\) Let's break down the problem step by step. ### Part 1: Solve the system of equations The first set of equations is: 1. \( x + 1 = b(b - x) \) 2. \( 4ax - 4bx = a^2 - 2ab + b^2 \) 3. \( \frac{x^2}{c^2} = b^2 \) 4. \( \frac{x}{a} - 3 = 5 - \frac{x}{b} \) 5. \( \frac{x + a}{x - b} = \frac{c}{3} \) 6. \( \frac{a}{x - 1} = \frac{b}{x + 2} \) 7. \( \frac{2x}{x + a} = \frac{b}{x - 1} + 2 \) Unfortunately, this set of equations is too complex to solve within the time limit, resulting in a timeout. Therefore, we cannot provide a solution for this part. ### Part 2: Solve the inequalities The second set of inequalities is: 1. \( 2x + 1 \geq 13 \) 2. \( 2 - 2x \leq 3 \) 3. \( 3(x + 7) > 6 \) 4. \( 10 - 2x < 22 + 2x \) 5. \( \frac{1 - x}{2} \leq 3 - x \) 6. \( \frac{-5x}{3} - 1 \geq x + 7 \) The results for this set of inequalities indicate that there are no solutions: - **Result**: \( x \in \varnothing \) ### Part 3: Solve the second set of inequalities The third set of inequalities is: 1. \( -4 \leq -2x < 6 \) 2. \( -8 \leq 4x - 12 < 0 \) 3. \( -7 \leq 3x - 1 < 5 \) 4. \( -2 \leq 4 - 2x \leq 8 \) 5. \( -4 < \frac{3x - 5}{2} < 8 \) 6. \( -5 < \frac{1 - 2x}{3} \leq 3 \) The results for this set of inequalities indicate: - **Result**: \( 1 \leq x < 2 \) ### Summary of Results 1. **First set of equations**: No solution found due to complexity. 2. **Second set of inequalities**: No solutions (\( x \in \varnothing \)). 3. **Third set of inequalities**: \( 1 \leq x < 2 \) ### Number Line Representation For the third set of inequalities, the solution can be represented on a number line as follows: - A closed circle at \( 1 \) (indicating \( 1 \) is included). - An open circle at \( 2 \) (indicating \( 2 \) is not included). This means the solution is the interval \([1, 2)\).