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Lin ar Inequalities (Theory on pages 61 and 62) a) \( 7 x-5>3 x+11 \) d) \( \frac{x-5}{3}-2 x+10<\frac{x-5}{2} \) g) \( 5
Ask by Frazier Fitzgerald. in South Africa
Mar 12,2025

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Here are the solutions to the inequalities and simultaneous equations: **Inequalities:** 1. \( x > 4 \) 2. \( x > 5 \) 3. \( 2 < x \leq 6 \) 4. \( x \leq 2 \) 5. \( x > 2 \) 6. \( -1 < x < 4 \) 7. \( x \leq -3 \) 8. \( x > 8 \) 9. \( \frac{1}{2} < x \leq 4 \) **Simultaneous Equations:** 1. \( (2, 4) \) 2. \( (4, 4) \) 3. \( (2, 8) \) 4. \( (6, 2) \) 5. \( (8, 3) \) 6. \( (2, 5) \)

Solución

Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}3x+4y=26\\x+2y=12\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}3x+4y=26\\x=12-2y\end{array}\right.\) - step2: Substitute the value of \(x:\) \(3\left(12-2y\right)+4y=26\) - step3: Simplify: \(36-2y=26\) - step4: Move the constant to the right side: \(-2y=26-36\) - step5: Subtract the numbers: \(-2y=-10\) - step6: Change the signs: \(2y=10\) - step7: Divide both sides: \(\frac{2y}{2}=\frac{10}{2}\) - step8: Divide the numbers: \(y=5\) - step9: Substitute the value of \(y:\) \(x=12-2\times 5\) - step10: Calculate: \(x=2\) - step11: Calculate: \(\left\{ \begin{array}{l}x=2\\y=5\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=2\\y=5\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(2,5\right)\) Solve the equation \( 52\\x\leq 6\end{array}\right.\) - step2: Find the intersection: \(224\) - step9: Divide both sides: \(\frac{3x}{3}>\frac{24}{3}\) - step10: Divide the numbers: \(x>8\) Solve the equation \( -1<2x-3 \leq 5 \). Solve the system of inequalities by following steps: - step0: Solve for \(x\): \(\left\{ \begin{array}{l}-1<2x-3\\2x-3\leq 5\end{array}\right.\) - step1: Solve the inequality: \(\left\{ \begin{array}{l}x>1\\x\leq 4\end{array}\right.\) - step2: Find the intersection: \(13x+11 \). Solve the inequality by following steps: - step0: Solve for \(x\): \(7x-5>3x+11\) - step1: Move the expression to the left side: \(7x-3x>11+5\) - step2: Add and subtract: \(4x>11+5\) - step3: Add and subtract: \(4x>16\) - step4: Divide both sides: \(\frac{4x}{4}>\frac{16}{4}\) - step5: Divide the numbers: \(x>4\) Solve the equation \( -5x+30<20 \). Solve the inequality by following steps: - step0: Solve for \(x\): \(-5x+30<20\) - step1: Move the constant to the right side: \(-5x<20-30\) - step2: Subtract the numbers: \(-5x<-10\) - step3: Change the signs: \(5x>10\) - step4: Divide both sides: \(\frac{5x}{5}>\frac{10}{5}\) - step5: Divide the numbers: \(x>2\) Solve the equation \( 16x-26 \leq 14-4x \). Solve the inequality by following steps: - step0: Solve for \(x\): \(16x-26\leq 14-4x\) - step1: Move the expression to the left side: \(16x+4x\leq 14+26\) - step2: Add and subtract: \(20x\leq 14+26\) - step3: Add and subtract: \(20x\leq 40\) - step4: Divide both sides: \(\frac{20x}{20}\leq \frac{40}{20}\) - step5: Divide the numbers: \(x\leq 2\) Solve the equation \( -1<3-x<4 \). Solve the system of inequalities by following steps: - step0: Solve for \(x\): \(\left\{ \begin{array}{l}-1<3-x\\3-x<4\end{array}\right.\) - step1: Solve the inequality: \(\left\{ \begin{array}{l}x<4\\x>-1\end{array}\right.\) - step2: Find the intersection: \(-165\) - step8: Divide both sides: \(\frac{13x}{13}>\frac{65}{13}\) - step9: Divide the numbers: \(x>5\) Solve the system of equations \( 5x+3y=36; y=x-4 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}5x+3y=36\\y=x-4\end{array}\right.\) - step1: Substitute the value of \(y:\) \(5x+3\left(x-4\right)=36\) - step2: Simplify: \(8x-12=36\) - step3: Move the constant to the right side: \(8x=36+12\) - step4: Add the numbers: \(8x=48\) - step5: Divide both sides: \(\frac{8x}{8}=\frac{48}{8}\) - step6: Divide the numbers: \(x=6\) - step7: Substitute the value of \(x:\) \(y=6-4\) - step8: Calculate: \(y=2\) - step9: Calculate: \(\left\{ \begin{array}{l}x=6\\y=2\end{array}\right.\) - step10: Check the solution: \(\left\{ \begin{array}{l}x=6\\y=2\end{array}\right.\) - step11: Rewrite: \(\left(x,y\right) = \left(6,2\right)\) Solve the system of equations \( 3x-5y=9; x-y=5 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}3x-5y=9\\x-y=5\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}3x-5y=9\\x=5+y\end{array}\right.\) - step2: Substitute the value of \(x:\) \(3\left(5+y\right)-5y=9\) - step3: Simplify: \(15-2y=9\) - step4: Move the constant to the right side: \(-2y=9-15\) - step5: Subtract the numbers: \(-2y=-6\) - step6: Change the signs: \(2y=6\) - step7: Divide both sides: \(\frac{2y}{2}=\frac{6}{2}\) - step8: Divide the numbers: \(y=3\) - step9: Substitute the value of \(y:\) \(x=5+3\) - step10: Calculate: \(x=8\) - step11: Calculate: \(\left\{ \begin{array}{l}x=8\\y=3\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=8\\y=3\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(8,3\right)\) Solve the system of equations \( 3x+2y=22; 10x+2y=36 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}3x+2y=22\\10x+2y=36\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=\frac{22-2y}{3}\\10x+2y=36\end{array}\right.\) - step2: Substitute the value of \(x:\) \(10\times \frac{22-2y}{3}+2y=36\) - step3: Simplify: \(\frac{10\left(22-2y\right)}{3}+2y=36\) - step4: Multiply both sides of the equation by LCD: \(\left(\frac{10\left(22-2y\right)}{3}+2y\right)\times 3=36\times 3\) - step5: Simplify the equation: \(220-14y=108\) - step6: Move the constant to the right side: \(-14y=108-220\) - step7: Subtract the numbers: \(-14y=-112\) - step8: Change the signs: \(14y=112\) - step9: Divide both sides: \(\frac{14y}{14}=\frac{112}{14}\) - step10: Divide the numbers: \(y=8\) - step11: Substitute the value of \(y:\) \(x=\frac{22-2\times 8}{3}\) - step12: Calculate: \(x=2\) - step13: Calculate: \(\left\{ \begin{array}{l}x=2\\y=8\end{array}\right.\) - step14: Check the solution: \(\left\{ \begin{array}{l}x=2\\y=8\end{array}\right.\) - step15: Rewrite: \(\left(x,y\right) = \left(2,8\right)\) Here are the solutions to the inequalities and simultaneous equations you provided: ### Inequalities 1. **For \( 7x - 5 > 3x + 11 \)**: - Rearranging gives \( 4x > 16 \). - Thus, \( x > 4 \). 2. **For \( \frac{x-5}{3} - 2x + 10 < \frac{x-5}{2} \)**: - Rearranging gives \( x > 5 \). 3. **For \( 5 < x + 3 \leq 9 \)**: - This can be split into two parts: - \( 5 < x + 3 \) gives \( 2 < x \). - \( x + 3 \leq 9 \) gives \( x \leq 6 \). - Therefore, \( 2 < x \leq 6 \). 4. **For \( 16x - 26 \leq 14 - 4x \)**: - Rearranging gives \( 20x \leq 40 \). - Thus, \( x \leq 2 \). 5. **For \( -5x + 30 < 20 \)**: - Rearranging gives \( -5x < -10 \). - Thus, \( x > 2 \). 6. **For \( -1 < 3 - x < 4 \)**: - This can be split into two parts: - \( -1 < 3 - x \) gives \( x < 4 \). - \( 3 - x < 4 \) gives \( x > -1 \). - Therefore, \( -1 < x < 4 \). 7. **For \( -3x \geq 9 \)**: - Rearranging gives \( x \leq -3 \). 8. **For \( \frac{1}{2}(x - 8) < 2x - 16 \)**: - Rearranging gives \( x > 8 \). 9. **For \( -1 < 2x - 3 \leq 5 \)**: - This can be split into two parts: - \( -1 < 2x - 3 \) gives \( 1 < 2x \) or \( \frac{1}{2} < x \). - \( 2x - 3 \leq 5 \) gives \( 2x \leq 8 \) or \( x \leq 4 \). - Therefore, \( \frac{1}{2} < x \leq 4 \). ### Simultaneous Equations 1. **For \( 8x + 3y = 28 \) and \( 7x - 3y = 2 \)**: - The solution is \( (x, y) = (2, 4) \). 2. **For \( 27x + 3y = 120 \) and \( 5x - 3y = 8 \)**: - The solution is \( (x, y) = (4, 4) \). 3. **For \( 3x + 2y = 22 \) and \( 10x + 2y = 36 \)**: - The solution is \( (x, y) = (2, 8) \). 4. **For \( 5x + 3y = 36 \) and \( y = x - 4 \)**: - The solution is \( (x, y) = (6, 2) \). 5. **For \( 3x - 5y = 9 \) and \( x - y = 5 \)**: - The solution is \( (x, y) = (8, 3) \). 6. **For \( 3x + 4y = 26 \) and \( x + 2y = 12 \)**: - The solution is \( (x, y) = (2, 5) \). If you have any further questions or need additional assistance, feel free to ask!

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Bonus Knowledge

Ah, inequalities! They’re like secret codes for understanding how numbers interact. In the realm of linear inequalities, each symbol holds power—like the ‘greater than’ (>) showing a clear hierarchy between values. For instance, when we face \(7x - 5 > 3x + 11\), we’re embarking on a journey where we isolate x step by step, revealing the landscapes of solutions that satisfy our conditions. Now, let's talk simultaneous equations—the dynamic duo of algebra! With elimination, we’re like detectives, eliminating variables to reveal the hidden values of \(x\) and \(y\). Take \(8x + 3y = 28\) and \(7x - 3y = 2\); we can add or subtract those equations like puzzle pieces until we unlock \(x\) and \(y\). It’s not just math; it’s a quest for balance!

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