To determine which integer from the set [tex]\( S = \{-1, 1, 3, 4\} \)[/tex] will make the equation [tex]\( 3p + 13 = 22 \)[/tex] true, let’s go through each step in detail.
Step 1: Start with the equation and isolate [tex]\( p \)[/tex]:
[tex]\[
3p + 13 = 22
\][/tex]
Step 2: Subtract 13 from both sides of the equation to isolate the term with [tex]\( p \)[/tex]:
[tex]\[
3p + 13 - 13 = 22 - 13
\][/tex]
This simplifies to:
[tex]\[
3p = 9
\][/tex]
Step 3: Divide both sides by 3 to solve for [tex]\( p \)[/tex]:
[tex]\[
p = \frac{9}{3} = 3
\][/tex]
Step 4: Now, we need to check the values in the given set [tex]\( S = \{-1, 1, 3, 4\} \)[/tex].
Check each element:
- For [tex]\( p = -1 \)[/tex]:
[tex]\[
3(-1) + 13 = -3 + 13 = 10 \quad (\text{not equal to } 22)
\][/tex]
- For [tex]\( p = 1 \)[/tex]:
[tex]\[
3(1) + 13 = 3 + 13 = 16 \quad (\text{not equal to } 22)
\][/tex]
- For [tex]\( p = 3 \)[/tex]:
[tex]\[
3(3) + 13 = 9 + 13 = 22 \quad (\text{equal to } 22)
\][/tex]
- For [tex]\( p = 4 \)[/tex]:
[tex]\[
3(4) + 13 = 12 + 13 = 25 \quad (\text{not equal to } 22)
\][/tex]
Upon checking each value in [tex]\( S \)[/tex], we see that the only integer that satisfies the equation [tex]\( 3p + 13 = 22 \)[/tex] is [tex]\( p = 3 \)[/tex].
Therefore, the integer in the solution set that makes the equation true is [tex]\( 3 \)[/tex].
