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

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 \( (x, y) = (2, 5) \). 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