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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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