Question

determining which equations have no solution
which of these equations have no solution? check all that apply.
□ $2(x + 2) + 2 = 2(x + 3) + 1$
□ $2x + 3(x + 5) = 5(x - 3)$
□ $4(x + 3) = x + 12$
□ $4 - (2x + 9) = \frac{1}{2}(- 4x - 2)$
□ $5(x + 4) - x = 4(x + 5) - 1$

Explanation:

Step1: Simplify first equation

Expand and simplify:
$2(x+2)+2 = 2(x+3)+1$
$2x + 4 + 2 = 2x + 6 + 1$
$2x + 6 = 2x + 7$
Subtract $2x$ from both sides: $6=7$ (false)

Step2: Simplify second equation

Expand and simplify:
$2x + 3(x+5) = 5(x-3)$
$2x + 3x + 15 = 5x - 15$
$5x + 15 = 5x - 15$
Subtract $5x$ from both sides: $15=-15$ (false)

Step3: Simplify third equation

Expand and simplify:
$4(x+3) = x + 12$
$4x + 12 = x + 12$
Subtract $x$ and 12: $3x=0$ → $x=0$ (has solution)

Step4: Simplify fourth equation

Expand and simplify:
$4-(2x+5) = \frac{1}{2}(-4x-2)$
$4-2x-5 = -2x -1$
$-2x -1 = -2x -1$
This is always true (infinitely many solutions)

Step5: Simplify fifth equation

Expand and simplify:
$5(x+4)-x = 4(x+5)-1$
$5x + 20 - x = 4x + 20 -1$
$4x + 20 = 4x + 19$
Subtract $4x$ from both sides: $20=19$ (false)

Answer:

1. $2(x + 2) + 2 = 2(x + 3) + 1$
2. $2x + 3(x + 5) = 5(x - 3)$
3. $5(x + 4) - x = 4(x + 5) - 1$