Question

Examine the equation.

        4(x – 3) = 4x – 12

Which of the following is true? (Check all that apply.)
It is a true statement.
Any input will result in an equivalent equation.
It is equivalent to an equation of the form a = a.
It has no solution.
Only one input will result in a true statement.

Answer 1

Let's examine the equation given:

\[ 4(x - 3) = 4x - 12 \]

First, let's simplify the left side:

\[ 4(x - 3) = 4x - 12 \]

Here, both sides of the equation are equal since we can expand the left side:

- Expanding the left side:
\[ 4(x - 3) = 4x - 12 \]
This simply confirms that \( 4(x - 3) \) and \( 4x - 12 \) are equivalent expressions.

Since both sides are equal, we can conclude the following:

1. **It is a true statement.** (True, because both sides are equal.)
2. **Any input will result in an equivalent equation.** (True, when we substitute any value for \(x\), the left-hand side will equal the right-hand side.)
3. **It is equivalent to an equation of the form a = a.** (True, since you can rewrite it as \(4x - 12 = 4x - 12\), which is truly a = a.)
4. **It has no solution.** (False, since every input will give a true statement.)
5. **Only one input will result in a true statement.** (False, as any value for \(x\) will satisfy the equation.)

Based on this analysis, the correct statements are:

- It is a true statement.
- Any input will result in an equivalent equation.
- It is equivalent to an equation of the form a = a.