Lin ar Inequalities (Theory on pages 61 and 62) a) $$ 7 x-53 x+11 $$ d) $$ \frac{x-5}{3}-2 x+10

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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

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Solve the system of equations by following steps: - step0: Solve using the substitution method: $$\left\{ \begin{array}{l}3x+4y=26\x+2y=12\end{array} ight.$$ - step1: Solve the equation: $$\left\{ \begin{array}{l}3x+4y=26\x=12-2y\end{array} ight.$$ - step2: Substitute the value of $$x:$$ $$3\left(12-2y ight)+4y=26$$ - 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 $$y:$$ $$x=12-2 imes 5$$ - step10: Calculate: $$x=2$$ - step11: Calculate: $$\left\{ \begin{array}{l}x=2\y=5\end{array} ight.$$ - step12: Check the solution: $$\left\{ \begin{array}{l}x=2\y=5\end{array} ight.$$ - step13: Rewrite: $$\left(x,y ight) = \left(2,5 ight)$$ Solve the equation $$ 52\x\leq 6\end{array} ight.$$ - 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 $$x$$: $$\left\{ \begin{array}{l}-1<2x-3\2x-3\leq 5\end{array} ight.$$ - step1: Solve the inequality: $$\left\{ \begin{array}{l}x>1\x\leq 4\end{array} ight.$$ - 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 $$x$$: $$\left\{ \begin{array}{l}-1<3-x\3-x<4\end{array} ight.$$ - step1: Solve the inequality: $$\left\{ \begin{array}{l}x<4\x>-1\end{array} ight.$$ - 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: $$\left\{ \begin{array}{l}5x+3y=36\y=x-4\end{array} ight.$$ - step1: Substitute the value of $$y:$$ $$5x+3\left(x-4 ight)=36$$ - 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: $$\left\{ \begin{array}{l}x=6\y=2\end{array} ight.$$ - step10: Check the solution: $$\left\{ \begin{array}{l}x=6\y=2\end{array} ight.$$ - step11: Rewrite: $$\left(x,y ight) = \left(6,2 ight)$$ 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: $$\left\{ \begin{array}{l}3x-5y=9\x-y=5\end{array} ight.$$ - step1: Solve the equation: $$\left\{ \begin{array}{l}3x-5y=9\x=5+y\end{array} ight.$$ - step2: Substitute the value of $$x:$$ $$3\left(5+y ight)-5y=9$$ - 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: $$\left\{ \begin{array}{l}x=8\y=3\end{array} ight.$$ - step12: Check the solution: $$\left\{ \begin{array}{l}x=8\y=3\end{array} ight.$$ - step13: Rewrite: $$\left(x,y ight) = \left(8,3 ight)$$ 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: $$\left\{ \begin{array}{l}3x+2y=22\10x+2y=36\end{array} ight.$$ - step1: Solve the equation: $$\left\{ \begin{array}{l}x=\frac{22-2y}{3}\10x+2y=36\end{array} ight.$$ - step2: Substitute the value of $$x:$$ $$10 imes \frac{22-2y}{3}+2y=36$$ - step3: Simplify: $$\frac{10\left(22-2y ight)}{3}+2y=36$$ - step4: Multiply both sides of the equation by LCD: $$\left(\frac{10\left(22-2y ight)}{3}+2y ight) imes 3=36 imes 3$$ - 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 $$y:$$ $$x=\frac{22-2 imes 8}{3}$$ - step12: Calculate: $$x=2$$ - step13: Calculate: $$\left\{ \begin{array}{l}x=2\y=8\end{array} ight.$$ - step14: Check the solution: $$\left\{ \begin{array}{l}x=2\y=8\end{array} ight.$$ - step15: Rewrite: $$\left(x,y ight) = \left(2,8 ight)$$ 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! 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. $$ (2, 5) $$

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

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Mar 12,2025

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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