Question

1. ( 8q + 12 = 4(3 + 2q) )
2. ( \frac{1}{2}(x + 6) = \frac{1}{2}x - 9 )
3. ( 12y + 48 - 4y = 8(y - 6) )
4. ( 8(z + 6) = 4(2z + 12) )

Explanation:

Problem 11: \( 8q + 12 = 4(3 + 2q) \)

Step 1: Expand the right side

Using the distributive property \( a(b + c)=ab + ac \), we get \( 4(3 + 2q)=12 + 8q \). So the equation becomes \( 8q + 12 = 12 + 8q \).

Step 2: Subtract \( 8q \) from both sides

Subtracting \( 8q \) from both sides gives \( 8q - 8q+ 12 = 12 + 8q - 8q \), which simplifies to \( 12 = 12 \).

Step 1: Expand the left side

Using the distributive property \( \frac{1}{2}(x + 6)=\frac{1}{2}x+\frac{1}{2}\times6=\frac{1}{2}x + 3 \). The equation is now \( \frac{1}{2}x + 3=\frac{1}{2}x - 9 \).

Step 2: Subtract \( \frac{1}{2}x \) from both sides

Subtracting \( \frac{1}{2}x \) from both sides gives \( \frac{1}{2}x-\frac{1}{2}x + 3=\frac{1}{2}x-\frac{1}{2}x - 9 \), which simplifies to \( 3=-9 \).

Step 1: Simplify the left side

Combine like terms: \( 12y-4y + 48 = 8y + 48 \). The equation becomes \( 8y + 48 = 8(y - 6) \).

Step 2: Expand the right side

Using the distributive property, \( 8(y - 6)=8y-48 \). So we have \( 8y + 48 = 8y - 48 \).

Step 3: Subtract \( 8y \) from both sides

Subtracting \( 8y \) from both sides gives \( 8y - 8y+ 48 = 8y - 8y - 48 \), which simplifies to \( 48=-48 \).

Answer:

This equation is an identity, meaning it is true for all real values of \( q \).

Problem 12: \( \frac{1}{2}(x + 6)=\frac{1}{2}x - 9 \)