To solve the equation [tex]\(\frac{p+1}{3} = \frac{p+1}{4}\)[/tex], we need to find the value of [tex]\(p\)[/tex] that satisfies this equality. Let's go through this step-by-step.
1. Write down the equation:
[tex]\[
\frac{p+1}{3} = \frac{p+1}{4}
\][/tex]
2. Eliminate the fractions by finding a common denominator, which in this case is 12. Multiply every term by 12:
[tex]\[
12 \cdot \frac{p+1}{3} = 12 \cdot \frac{p+1}{4}
\][/tex]
3. Simplify both sides:
[tex]\[
4(p+1) = 3(p+1)
\][/tex]
4. Distribute the constants on both sides:
[tex]\[
4p + 4 = 3p + 3
\][/tex]
5. Bring all terms involving [tex]\(p\)[/tex] to one side and constants to the opposite side. Subtract [tex]\(3p\)[/tex] and [tex]\(3\)[/tex] from both sides:
[tex]\[
4p + 4 - 3p - 3 = 0
\][/tex]
[tex]\[
p + 1 = 0
\][/tex]
6. Solve for [tex]\(p\)[/tex]:
[tex]\[
p = -1
\][/tex]
7. Verify the solution: To ensure [tex]\(p = -1\)[/tex] is correct, substitute [tex]\(p = -1\)[/tex] back into the original equation:
[tex]\[
\frac{(-1)+1}{3} = \frac{(-1)+1}{4}
\][/tex]
[tex]\[
\frac{0}{3} = \frac{0}{4}
\][/tex]
[tex]\[
0 = 0
\][/tex]
Since both sides of the equation are equal when [tex]\(p = -1\)[/tex], this value satisfies the equation.
Therefore, the value of [tex]\(p\)[/tex] that makes the equation true is:
[tex]\[
\boxed{-1}
\][/tex]
