To determine which integer in the solution set [tex]\( S = \{-2, 0, 7, 9\} \)[/tex] will make the equation true, we need to test each integer by substituting it into the given equation:
[tex]\[ 6x - 8 = 2(2x + 3) \][/tex]
Let's start by simplifying the given equation:
Step 1: Expand the right-hand side:
[tex]\[ 6x - 8 = 4x + 6 \][/tex]
Step 2: Move all terms involving [tex]\( x \)[/tex] to one side of the equation and constant terms to the other side:
[tex]\[ 6x - 4x - 8 = 6 \][/tex]
[tex]\[ 2x - 8 = 6 \][/tex]
Step 3: Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x = 6 + 8 \][/tex]
[tex]\[ 2x = 14 \][/tex]
[tex]\[ x = 7 \][/tex]
Now we know that if [tex]\( x = 7 \)[/tex], the equation holds true. We'll verify this with the provided solution set [tex]\( S \)[/tex] to ensure only 7 satisfies the equation:
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[ 6(-2) - 8 = 2(2(-2) + 3) \][/tex]
[tex]\[ -12 - 8 = 2(-4 + 3) \][/tex]
[tex]\[ -20 = 2(-1) \][/tex]
[tex]\[ -20 \neq -2 \][/tex]
2. For [tex]\( x = 0 \)[/tex]:
[tex]\[ 6(0) - 8 = 2(2(0) + 3) \][/tex]
[tex]\[ 0 - 8 = 2(0 + 3) \][/tex]
[tex]\[ -8 = 2(3) \][/tex]
[tex]\[ -8 \neq 6 \][/tex]
3. For [tex]\( x = 7 \)[/tex]:
[tex]\[ 6(7) - 8 = 2(2(7) + 3) \][/tex]
[tex]\[ 42 - 8 = 2(14 + 3) \][/tex]
[tex]\[ 34 = 2(17) \][/tex]
[tex]\[ 34 = 34 \][/tex]
4. For [tex]\( x = 9 \)[/tex]:
[tex]\[ 6(9) - 8 = 2(2(9) + 3) \][/tex]
[tex]\[ 54 - 8 = 2(18 + 3) \][/tex]
[tex]\[ 46 = 2(21) \][/tex]
[tex]\[ 46 \neq 42 \][/tex]
Among the integers in the solution set [tex]\( S \)[/tex], the integer [tex]\( 7 \)[/tex] is the only value that satisfies the given equation.
Therefore, the integer in the solution set that makes the equation [tex]\( 6x - 8 = 2(2x + 3) \)[/tex] true is [tex]\( \boxed{7} \)[/tex].
