Question

3.11: special cases
use notebook paper. you must write the original equation on your paper.
solve each equation.
work must be shown going down under the original equation. each step is written under the previous step. each line may contain only one equal sign.
you must show inverse operations on both sides of the equation.
show all work! solutions only receive no credit.

1. \\(7x + 2(x - 2) = 3(3x + 4)\\) 2. \\(-5(x - 1) + x = 6x - 11\\)
2. \\(3(x - 1) - 2x = -2x + 3(x - 1)\\) 4. \\(4(x - 2) - 2x = 3x - (x - 2)\\)
3. \\(4x - 1 - 2(x + 1) = 3x - (4x + 2)\\) 6. \\(8x + 38 = -3(-6 - 4x)\\)
4. \\(5x + 3 = 5(x + 2) - 7\\) 8. \\(5(x + 6) - 2x = 3(x + 10)\\)
5. \\(8(x + 2) = 2x + 16\\) 10. \\(2y - 3 = 3y - 2 - y - 1\\)

Explanation:

Step1: Solve Equation 1: \(7x + 2(x - 2)=3(3x + 4)\)

First, expand the parentheses. For the left side, \(2(x - 2)=2x-4\), so left side becomes \(7x + 2x-4 = 9x-4\). For the right side, \(3(3x + 4)=9x + 12\). Now the equation is \(9x-4=9x + 12\). Subtract \(9x\) from both sides: \(9x-9x-4=9x-9x + 12\), which simplifies to \(-4 = 12\). This is a contradiction, so no solution.

Step2: Solve Equation 2: \(-5(x - 1)+x=6x - 11\)

Expand the left side: \(-5x + 5+x=-4x + 5\). Now the equation is \(-4x + 5=6x - 11\). Add \(4x\) to both sides: \(-4x+4x + 5=6x + 4x-11\), so \(5 = 10x-11\). Add 11 to both sides: \(5 + 11=10x-11 + 11\), \(16 = 10x\). Divide by 10: \(x=\frac{16}{10}=\frac{8}{5}\).

Step3: Solve Equation 3: \(3(x - 1)-2x=-2x + 3(x - 1)\)

Expand left side: \(3x-3-2x=x - 3\). Right side: \(-2x+3x - 3=x - 3\). Now the equation is \(x - 3=x - 3\). Subtract \(x\) from both sides: \(x-x - 3=x-x - 3\), which gives \(-3=-3\). This is an identity, so all real numbers are solutions.

Step4: Solve Equation 4: \(4(x - 2)-2x=3x-(x - 2)\)

Expand left side: \(4x-8-2x=2x-8\). Right side: \(3x-x + 2=2x + 2\). Equation: \(2x-8=2x + 2\). Subtract \(2x\): \(2x-2x-8=2x-2x + 2\), \(-8 = 2\). Contradiction, no solution.

Step5: Solve Equation 5: \(4x-1-2(x + 1)=3x-(4x + 2)\)

Expand left side: \(4x-1-2x-2=2x-3\). Right side: \(3x-4x-2=-x - 2\). Equation: \(2x-3=-x - 2\). Add \(x\): \(2x+x-3=-x+x - 2\), \(3x-3=-2\). Add 3: \(3x-3 + 3=-2 + 3\), \(3x = 1\). Divide by 3: \(x=\frac{1}{3}\).

Step6: Solve Equation 6: \(8x + 38=-3(-6 - 4x)\)

Expand right side: \(-3\times(-6)-3\times(-4x)=18 + 12x\). Equation: \(8x + 38=12x + 18\). Subtract \(8x\): \(8x-8x + 38=12x-8x + 18\), \(38 = 4x + 18\). Subtract 18: \(38-18=4x + 18-18\), \(20 = 4x\). Divide by 4: \(x = 5\).

Step7: Solve Equation 7: \(5x + 3=5(x + 2)-7\)

Expand right side: \(5x + 10-7=5x + 3\). Equation: \(5x + 3=5x + 3\). Subtract \(5x\): \(5x-5x + 3=5x-5x + 3\), \(3 = 3\). Identity, all real numbers.

Step8: Solve Equation 8: \(5(x + 6)-2x=3(x + 10)\)

Expand left side: \(5x + 30-2x=3x + 30\). Right side: \(3x + 30\). Equation: \(3x + 30=3x + 30\). Subtract \(3x\): \(3x-3x + 30=3x-3x + 30\), \(30 = 30\). Identity, all real numbers.

Step9: Solve Equation 9: \(8(x + 2)=2x + 16\)

Expand left side: \(8x + 16=2x + 16\). Subtract \(2x\): \(8x-2x + 16=2x-2x + 16\), \(6x + 16=16\). Subtract 16: \(6x + 16-16=16-16\), \(6x = 0\). Divide by 6: \(x = 0\).

Step10: Solve Equation 10: \(2y-3=3y-2-y-1\)

Simplify right side: \(3y-y-2-1=2y-3\). Equation: \(2y-3=2y-3\). Subtract \(2y\): \(2y-2y-3=2y-2y-3\), \(-3=-3\). Identity, all real numbers.

Answer:

1. No solution
2. \(x=\frac{8}{5}\)
3. All real numbers
4. No solution
5. \(x=\frac{1}{3}\)
6. \(x = 5\)
7. All real numbers
8. All real numbers
9. \(x = 0\)
10. All real numbers