Question

To solve the equation, Antwan uses the distributive property, combines like terms, then uses the addition and subtraction properties of equality to isolate the variable term on one side of the equation and the constant term on the other side. What are the possible values of the constant after Antwan has completed these steps?

3 (one-third x + 1) + 4 = negative 1 minus 4 (x + 3)
–3 or 3
–3 or 5
–20 or 5
–20 or 20

Answer 1

Let's solve the equation step by step according to the method described:

The given equation is:

\[ 3\left(\frac{1}{3}x + 1\right) + 4 = -1 - 4(x + 3) \]

### Step 1: Apply the distributive property

First, distribute on both sides of the equation.

Left-hand side:
\[ 3\left(\frac{1}{3}x\right) + 3(1) + 4 = x + 3 + 4 = x + 7 \]

Right-hand side:
\[ -1 - 4(x + 3) = -1 - 4x - 12 = -1 - 12 - 4x = -13 - 4x \]

Now the equation is:
\[ x + 7 = -13 - 4x \]

### Step 2: Combine like terms

Next, we'll isolate the variable term on one side and the constant term on the other side. We can add \( 4x \) to both sides and subtract \( 7 \) from both sides:

\[ x + 4x + 7 - 7 = -13 - 4x + 4x - 7 \]

This simplifies to:
\[ 5x = -20 \]

### Step 3: Solve for \( x \)

Now, divide both sides by 5:
\[ x = -4 \]

### Conclusion

After completing these steps, the constant term in regards to rearranging or modifying forms of the equation (if considering \( -20 \) as a constant derived from isolating \( x \)) is \( -20 \).

Thus, from the provided options, the answer is:

**–20 or 20**

However, since the original determination was concerning constants derived from behavior of the equation, \( -20 \) would be the sought value applicable in this context.