Question

which two statements are correct when simplifying this expression?
12 + 6(3x - 2)
a first, calculate 6(3x - 2)
b first, calculate 12 + 6.
c 12 + 6(3x - 2) = 54x - 36
d 12 + 6(3x - 2) = 18x
e 12 + 6(3x - 2) = 18x - 24

Explanation:

Step1: Analyze the order of operations

When simplifying an expression with parentheses, we use the distributive property first on the parentheses. So for \(12 + 6(3x - 2)\), we first calculate \(6(3x - 2)\) (by distributing the 6), which means option A is correct and option B is incorrect (since we don't add 12 and 6 first, we handle the parentheses first).

Step2: Simplify the expression

Using the distributive property: \(6(3x - 2)=6\times3x - 6\times2 = 18x - 12\). Then we add the 12 from the original expression: \(12 + 18x - 12\). The 12 and -12 cancel out, leaving \(18x\)? Wait, no, wait. Wait, let's do it again. Wait, the original expression is \(12 + 6(3x - 2)\). Distribute the 6: \(6\times3x = 18x\), \(6\times(-2)= -12\). So now we have \(12 + 18x - 12\). Now combine like terms: 12 - 12 is 0, so we get \(18x\)? Wait, but let's check the options. Wait, option D says \(12 + 6(3x - 2)=18x\), and let's check the other options. Wait, maybe I made a mistake. Wait, no, let's re-express:

Wait, \(6(3x - 2)=18x - 12\). Then \(12 + 18x - 12\). 12 - 12 is 0, so \(18x\). But wait, option E is \(18x - 24\), option C is \(54x - 36\). Wait, maybe my initial distribution was wrong? Wait, no, 6 times 3x is 18x, 6 times -2 is -12. Then 12 + 18x -12. The 12 and -12 cancel, so 18x. So option D is correct? But wait, let's check option A: first, calculate 6(3x - 2). That's correct because of order of operations (parentheses first, using distributive property). So A is correct. Then let's see the simplification. Let's do it step by step:

\(12 + 6(3x - 2)\)

First, distribute the 6: \(6\times3x = 18x\), \(6\times(-2)= -12\). So now we have \(12 + 18x - 12\). Now, 12 - 12 = 0, so we have \(18x\). So option D is correct? But wait, let's check the options again. Wait, maybe I messed up the distribution. Wait, no, 6(3x - 2) is 18x - 12. Then add 12: 12 + 18x -12 = 18x. So D is correct. And A is correct. Wait, but let's check the options. The question is which two statements are correct. So A and D? Wait, but let's check again. Wait, maybe I made a mistake. Wait, let's re-express the original expression:

Wait, \(12 + 6(3x - 2)\). Let's compute 6(3x - 2) first (option A is correct, because we use the distributive property on the parentheses, so we handle the 6(3x -2) first, not 12 +6). Then, simplifying:

\(6(3x - 2) = 18x - 12\)

Then \(12 + 18x - 12 = 18x\) (since 12 -12 =0). So option D is correct. So the two correct statements are A and D? Wait, but let's check the options again. Wait, maybe I made a mistake. Wait, let's check option E: \(18x -24\). How would that happen? If we did 12 +6(3x -2) =12 +18x -12? No, 6*(-2) is -12, so 12 +18x -12. So 18x. So D is correct. And A is correct. So the two correct statements are A and D.

Wait, but let's confirm with the options:

A: First, calculate 6(3x - 2) – correct (order of operations, handle parentheses first via distributive property).

B: First, calculate 12 + 6 – incorrect (we don't add 12 and 6 first, we handle the parentheses).

C: 12 + 6(3x -2)=54x -36 – incorrect (63x is 18x, not 54x; 6(-2) is -12, not -36).

D: 12 + 6(3x -2)=18x – correct (as shown above).

E: 12 + 6(3x -2)=18x -24 – incorrect (6*(-2) is -12, not -24; then 12 -12 is 0, not -24).

So the correct statements are A and D.

Answer:

A. First, calculate \(6(3x - 2)\), D. \(12 + 6(3x - 2) = 18x\)