Cool Down: Choose Your Own Solution $$ \begin{array}{l}3 x+8=3 x+ \ ext { What value could you write in after } 3 x ext { that would make the equation true for: } \ ext { 1. no values of } x ext { ? } \ ext { 2. all values of } x ext { ? } \ ext { 3. just one value of } x ext { ? }\end{array} $$ ?

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Cool Down: Choose Your Own Solution $$ \begin{array}{l}3 x+8=3 x+ \ 	ext { What value could you write in after } 3 x 	ext { that would make the equation true for: } \ 	ext { 1. no values of } x 	ext { ? } \ 	ext { 2. all values of } x 	ext { ? } \ 	ext { 3. just one value of } x 	ext { ? }\end{array} $$ ?

UpStudy AI · Accepted Answer

Certainly! Let's analyze the equation and determine appropriate values to place after $$ 3x $$ to satisfy each of the given conditions.

**Given Equation:**
$$
3x + 8 = 3x + \text{[value]}
$$

We need to choose a suitable **[value]** such that:

1. **No values of $$ x $$ satisfy the equation.**
2. **All values of $$ x $$ satisfy the equation.**
3. **Only one specific value of $$ x $$ satisfies the equation.**

Let's explore each scenario in detail.

---

### 1. **No Values of $$ x $$ Satisfy the Equation**

**Objective:** Make the equation **false** for all possible values of $$ x $$.

**Approach:**
- Substitute **[value]** with a **constant** that is **different** from the constant term on the left side.
- This creates a contradiction, making the equation unsolvable.

**Example:**
$$
3x + 8 = 3x + 5
$$
**Explanation:**
- Subtract $$ 3x $$ from both sides:
  $$
  8 = 5
  $$
- This is a false statement, meaning there's **no solution**.

**General Rule:**  
Set **[value]** to any constant **not equal** to 8.

---

### 2. **All Values of $$ x $$ Satisfy the Equation**

**Objective:** Make the equation **always true**, regardless of the value of $$ x $$.

**Approach:**
- Substitute **[value]** with the **same constant** that's on the left side.
- This ensures both sides of the equation are identical.

**Example:**
$$
3x + 8 = 3x + 8
$$
**Explanation:**
- Subtract $$ 3x $$ from both sides:
  $$
  8 = 8
  $$
- This is always true, meaning **all values** of $$ x $$ are solutions.

**General Rule:**  
Set **[value]** to **8**.

---

### 3. **Only One Specific Value of $$ x $$ Satisfies the Equation**

**Objective:** Make the equation **true for exactly one value** of $$ x $$.

**Approach:**
- Substitute **[value]** with an **expression involving $$ x $$**.
- This introduces an additional variable, allowing the equation to be solvable for a unique $$ x $$.

**Example:**
$$
3x + 8 = 3x + x + 5
$$
**Simplify and Solve:**
$$
3x + 8 = 4x + 5
$$
$$
8 - 5 = 4x - 3x
$$
$$
3 = x
$$
**Explanation:**
- The equation simplifies to $$ x = 3 $$, meaning **only $$ x = 3 $$** satisfies the equation.

**Another Example:**
$$
3x + 8 = 3x - 2x + 10
$$
**Simplify and Solve:**
$$
3x + 8 = x + 10
$$
$$
3x - x = 10 - 8
$$
$$
2x = 2 \Rightarrow x = 1
$$
**Explanation:**
- The equation simplifies to $$ x = 1 $$, meaning **only $$ x = 1 $$** satisfies the equation.

**General Rule:**  
Set **[value]** to an expression of the form **$$ ax + b $$**, where $$ a \neq 0 $$. This introduces a variable term, allowing the equation to have a unique solution.

---

### **Summary**

- **No Solution:**  
  $$\boxed{3x + 8 = 3x + 5}$$  
  *(Any constant not equal to 8)*

- **All Solutions:**  
  $$\boxed{3x + 8 = 3x + 8}$$  
  *(The constant 8)*

- **One Solution:**  
  $$\boxed{3x + 8 = 3x + x + 5}$$  
  *(An expression involving $$ x $$, such as $$ x + 5 $$)*

By appropriately choosing the **[value]**, you can control the nature of the solutions to the equation.
To make the equation $$3x + 8 = 3x + \text{[value]}$$ true: 1. **No values of $$ x $$:** Use any constant different from 8, e.g., $$3x + 8 = 3x + 5$$. 2. **All values of $$ x $$:** Use the same constant as on the left, e.g., $$3x + 8 = 3x + 8$$. 3. **One value of $$ x $$:** Use an expression involving $$ x $$, e.g., $$3x + 8 = 3x + x + 5$$.

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