whether it has no solutions, exactly one solution, or infinitely many solutions. if an equation has one solution, solve to find the value that makes the statement true. explain why you chose no solution or infinitely many solutions.
2
a. $\frac{1}{4}(12 - 4x) = 3 - x$

b. $x - 3 = 3 - x$

c. $x - 3 = 3 + x$
3.
a. $-5x - 3x + 2 = -8x + 2$

b. $-5x - 3x - 4 = -8x + 2$

c. $-5x - 4x - 2 = -8x + 2$
4.
a. $4(2x - 2) + 2 = 4(x - 2)$

b. $4x + 2(2x - 3) = 8(x - 1)$

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

b. $x - 3(2 + 3x) = 2(5x - 3)$

c. $x - 3(2 - 3x) = 2(5x - 3)$

Question

whether it has no solutions, exactly one solution, or infinitely many solutions. if an equation has one solution, solve to find the value that makes the statement true. explain why you chose no solution or infinitely many solutions.
2
  a. $\frac{1}{4}(12 - 4x) = 3 - x$

b. $x - 3 = 3 - x$

c. $x - 3 = 3 + x$
3.
  a. $-5x - 3x + 2 = -8x + 2$

b. $-5x - 3x - 4 = -8x + 2$

c. $-5x - 4x - 2 = -8x + 2$
4.
  a. $4(2x - 2) + 2 = 4(x - 2)$

b. $4x + 2(2x - 3) = 8(x - 1)$

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

b. $x - 3(2 + 3x) = 2(5x - 3)$

c. $x - 3(2 - 3x) = 2(5x - 3)$

Accepted Answer

### 2a Solution:
# Explanation:
## Step1: Distribute the fraction
Multiply $\frac{1}{4}$ with each term inside the parentheses: $\frac{1}{4}(12 - 4x)=\frac{12}{4}-\frac{4x}{4}=3 - x$.
## Step2: Analyze the equation
After simplifying, we get $3 - x = 3 - x$. This is an identity, meaning it holds true for all real values of $x$.
# Answer:
The equation $\frac{1}{4}(12 - 4x)=3 - x$ has infinitely many solutions because simplifying it results in an identity ($3 - x = 3 - x$) that is true for all $x$.

### 2b Solution:
# Explanation:
## Step1: Rearrange terms
Add $x$ to both sides and add $3$ to both sides: $x + x=3 + 3$.
## Step2: Solve for $x$
Simplify to get $2x = 6$, then divide both sides by $2$: $x = 3$.
# Answer:
The equation $x - 3 = 3 - x$ has exactly one solution, $x = 3$, because we can solve it to find a unique value for $x$.

### 2c Solution:
# Explanation:
## Step1: Rearrange terms
Subtract $x$ from both sides: $x - x-3 = 3 + x - x$.
## Step2: Analyze the result
Simplifying gives $-3 = 3$, which is a false statement.
# Answer:
The equation $x - 3 = 3 + x$ has no solutions because simplifying it leads to a false statement ($-3 = 3$).

### 3a Solution:
# Explanation:
## Step1: Combine like terms
Combine $-5x$ and $-3x$ on the left: $-8x + 2=-8x + 2$.
## Step2: Analyze the equation
This is an identity, so it holds for all $x$.
# Answer:
The equation $-5x - 3x + 2=-8x + 2$ has infinitely many solutions as it simplifies to an identity ($-8x + 2=-8x + 2$).

### 3b Solution:
# Explanation:
## Step1: Combine like terms
Combine $-5x$ and $-3x$ on the left: $-8x - 4=-8x + 2$.
## Step2: Analyze the equation
Add $8x$ to both sides: $-4 = 2$, which is false.
# Answer:
The equation $-5x - 3x - 4=-8x + 2$ has no solutions because simplifying leads to $-4 = 2$ (a false statement).

### 3c Solution:
# Explanation:
## Step1: Combine like terms
Combine $-5x$ and $-4x$ on the left: $-9x - 2=-8x + 2$.
## Step2: Solve for $x$
Add $9x$ to both sides: $-2 = x + 2$, then subtract $2$ from both sides: $x=-4$.
# Answer:
The equation $-5x - 4x - 2=-8x + 2$ has exactly one solution, $x = - 4$, as we can solve it to find a unique $x$.

### 4a Solution:
# Explanation:
## Step1: Distribute and simplify
Distribute $4$ on the left: $8x-8 + 2 = 4x-8$, then simplify to $8x-6 = 4x-8$.
## Step2: Solve for $x$
Subtract $4x$ from both sides: $4x-6=-8$, then add $6$ to both sides: $4x=-2$, and divide by $4$: $x =-\frac{1}{2}$.
# Answer:
The equation $4(2x - 2)+2 = 4(x - 2)$ has exactly one solution, $x =-\frac{1}{2}$, as we can find a unique $x$ by solving.

### 4b Solution:
# Explanation:
## Step1: Distribute and simplify
Distribute $2$ on the left and $8$ on the right: $4x + 4x-6 = 8x-8$, then simplify to $8x-6 = 8x-8$.
## Step2: Analyze the result
Subtract $8x$ from both sides: $-6=-8$, which is false.
# Answer:
The equation $4x + 2(2x - 3)=8(x - 1)$ has no solutions because simplifying leads to a false statement ($-6=-8$).

### 4c Solution:
# Explanation:
## Step1: Distribute and simplify
Distribute $2$ on the left and $4$ on the right: $4x + 4x-6 = 8x-8 + 2$, then simplify to $8x-6 = 8x-6$.
## Step2: Analyze the equation
This is an identity, so it holds for all $x$.
# Answer:
The equation $4x + 2(2x - 3)=4(2x - 2)+2$ has infinitely many solutions because it simplifies to an identity ($8x - 6 = 8x - 6$).

### 5a Solution:
# Explanation:
## Step1: Distribute the terms
Distribute $-3$ on the left and $2$ on the right: $x-6 + 9x = 10x+6$.
## Step2: Combine like terms and simplify
Combine $x$ and $9x$: $10x-6 = 10x+6$. Subtract $10x$ from both sides: $-6 = 6$, which is false.
# Answer:
The equation $x - 3(2 - 3x)=2(5x + 3)$ has no solutions because simplifying leads to a false statement ($-6 = 6$).

### 5b Solution:
# Explanation:
## Step1: Distribute the terms
Distribute $-3$ on the left and $2$ on the right: $x-6 - 9x = 10x-6$.
## Step2: Combine like terms and solve
Combine $x$ and $-9x$: $-8x-6 = 10x-6$. Add $8x$ to both sides and add $6$ to both sides: $0 = 18x$, so $x = 0$.
# Answer:
The equation $x - 3(2 + 3x)=2(5x - 3)$ has exactly one solution, $x = 0$, as we can solve it to find a unique $x$.

### 5c Solution:
# Explanation:
## Step1: Distribute the terms
Distribute $-3$ on the left and $2$ on the right: $x-6 + 9x = 10x-6$.
## Step2: Combine like terms and analyze
Combine $x$ and $9x$: $10x-6 = 10x-6$. This is an identity.
# Answer:
The equation $x - 3(2 - 3x)=2(5x - 3)$ has infinitely many solutions because it simplifies to an identity ($10x - 6 = 10x - 6$) that holds for all $x$.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

2a Solution:

Step1: Distribute the fraction

Step2: Analyze the equation

Step1: Rearrange terms

Step2: Solve for \(x\)

Step1: Rearrange terms

Step2: Analyze the result

Answer:

2b Solution: