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

Ask by O'Quinn Stuart. in South Africa
Mar 13,2025

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Tutor-Verified Answer

Answer

There are no solutions for the first set of equations and the second set of inequalities. For the third set of inequalities, the solution is \(1 \leq x < 2\).

Solution

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 \([1, 2)\).

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Did you know that solving equations and inequalities dates back to ancient civilizations? The Babylonians were among the first to systematically solve equations around 2000 BCE. They used geometric methods and developed a number system that allowed them to work with quadratic equations long before the advent of modern algebra. Fast forward to today, and we have a wide array of tools for solving complex problems with mere clicks! In the real world, the skills you develop from solving equations have practical applications everywhere—from budgeting personal finances to modeling scientific phenomena. Engineers, architects, and even video game designers rely on algebra to navigate balance, design, and probability. So, every time you calculate how much cash you need to save for a new gadget or determine the trajectory of a flying object, you're utilizing those equation-solving skills in a big way!

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