To identify which postulate is illustrated by the equation \(2(x + 3) = 2x + 6\), we first need to understand what each postulate signifies and see how they apply to our equation:
1. The commutative postulate for multiplication states that for any two real numbers \(a\) and \(b\), \(a \cdot b = b \cdot a\).
2. Multiplication identity states that multiplying any number by 1 yields that number, i.e., \(a \cdot 1 = a\).
3. The addition inverse postulate states that for any real number \(a\), there exists an additive inverse \(-a\) such that \(a + (-a) = 0\).
4. Commutative postulate for addition states that for any two real numbers \(a\) and \(b\), \(a + b = b + a\).
5. The distributive postulate states that for any three real numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\).
6. The addition of zero postulate states that adding zero to any number yields that number, i.e., \(a + 0 = a\).
7. The multiplication inverse states that for any non-zero real number \(a\), there exists a multiplicative inverse \(\frac{1}{a}\) such that \(a \cdot \frac{1}{a} = 1\).
Now, let’s apply this to the given equation:
[tex]\[2(x + 3) = 2x + 6\][/tex]
This equation shows that a single term, \(2\), is applied to each term inside the parentheses, \(x + 3\). This is consistent with the distributive property of multiplication over addition.
According to the distributive postulate:
[tex]\[a(b + c) = ab + ac\][/tex]
In the equation, \(a = 2\), \(b = x\), and \(c = 3\). When we apply the distributive property, we multiply \(2\) by both \(x\) and \(3\) which gives:
[tex]\[2 \cdot x + 2 \cdot 3 = 2x + 6\][/tex]
Thus, the postulate that is illustrated for the real numbers
[tex]\[2(x + 3) = 2x + 6\][/tex]
is the distributive postulate.
