Question

1. one solution, no solution, or infinite solutions? $2(x + 6) = 2x + 26$
2. one solution, no solution, or infinite solutions? $-(-12x + 21) = -6 + 12x$
3. one solution, no solution, or infinite solutions? $42 + 6x = 6(-1 + 5x)$
4. one solution, no solution, or infinite solutions? $12 - 17x = 12(-x + 9) - 5x$
5. one solution, no solution, or infinite solutions? $8 + 11x - 10 = 10x + x + 2$
6. one solution, no solution, or infinite solutions? $18x - 2 = 2(9x -1)$
7. one solution, no solution, or infinite solutions? $-8(x - 1) = 8 - 8x$
8. one solution, no solution, or infinite solutions? $6(x -1) = 4x + 18$

Explanation:

Problem 2

Step1: Expand left side

$2(x+6)=2x+12$

Step2: Rearrange to isolate x

$2x+12=2x+26$
$2x-2x=26-12$
$0=14$

Step3: Interpret result

False statement, no solution.

Problem 3

Step1: Simplify left side

$-(-12x+21)=12x-21$

Step2: Rearrange to isolate x

$12x-21=-6+12x$
$12x-12x=-6+21$
$0=15$

Step3: Interpret result

False statement, no solution.

Problem 5

Step1: Expand right side

$6(-1+5x)=-6+30x$

Step2: Rearrange to isolate x

$42+6x=-6+30x$
$42+6=30x-6x$
$48=24x$

Step3: Solve for x

$x=\frac{48}{24}=2$

Step4: Interpret result

Single valid x, one solution.

Problem 6

Step1: Expand right side

$12(-x+9)-5x=-12x+108-5x=-17x+108$

Step2: Rearrange to isolate x

$12-17x=-17x+108$
$-17x+17x=108-12$
$0=96$

Step3: Interpret result

False statement, no solution.

Problem 8

Step1: Simplify right side

$10x+x+2=11x+2$

Step2: Rearrange to isolate x

$8+11x-10=11x+2$
$11x-2=11x+2$
$11x-11x=2+2$
$0=4$

Step3: Interpret result

False statement, no solution.

Problem 9

Step1: Expand right side

$2(9x-1)=18x-2$

Step2: Rearrange to isolate x

$18x-2=18x-2$
$18x-18x=-2+2$
$0=0$

Step3: Interpret result

True statement, infinite solutions.

Problem 11

Step1: Expand left side

$-8(x-1)=-8x+8$

Step2: Rearrange to isolate x

$-8x+8=8-8x$
$-8x+8x=8-8$
$0=0$

Step3: Interpret result

True statement, infinite solutions.

Problem 12

Step1: Expand left side

$6(x-1)=6x-6$

Step2: Rearrange to isolate x

$6x-6=4x+18$
$6x-4x=18+6$
$2x=24$

Step3: Solve for x

$x=\frac{24}{2}=12$

Step4: Interpret result

Single valid x, one solution.

Answer:

1. Problem 2: No Solution
2. Problem 3: No Solution
3. Problem 5: One Solution ($x=2$)
4. Problem 6: No Solution
5. Problem 8: No Solution
6. Problem 9: Infinite Solutions
7. Problem 11: Infinite Solutions
8. Problem 12: One Solution ($x=12$)