Question

solve the following absolute value equation.

\\|2p + 9| = |3p - 4|\\

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the solution set is \\{\quad\\}.
(simplify your answer. use a comma to separate answers as needed.)

b. the solution set is \\{\\} or \varnothing.

Explanation:

⚡ Using what you learned: Solving Absolute Value Equations

Step 1: Set up the two cases

An equation of the form \( |a| = |b| \) is true if the expressions inside are either equal or opposites:

$$ 2p + 9 = 3p - 4 \quad \text{or} \quad 2p + 9 = -(3p - 4) $$

Step 2: Solve Case 1

$$ 2p + 9 = 3p - 4 $$

$$ 9 + 4 = 3p - 2p $$

$$ p = 13 $$

Step 3: Solve Case 2

$$ 2p + 9 = -3p + 4 $$

$$ 5p + 9 = 4 $$

$$ 5p = -5 $$

$$ p = -1 $$

Answer:

A. The solution set is \( \{13, -1\} \).