Question

solve the following absolute value equation.

\\|2p + 19| = |3p - 9|\\

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:

$$ 2p + 19 = 3p - 9 \quad \text{or} \quad 2p + 19 = -(3p - 9) $$

Step 2: Solve Case 1

$$ 2p + 19 = 3p - 9 $$

$$ 19 + 9 = 3p - 2p $$

$$ p = 28 $$

Step 3: Solve Case 2

$$ 2p + 19 = -3p + 9 $$

$$ 2p + 3p = 9 - 19 $$

$$ 5p = -10 $$

$$ p = -2 $$

Answer:

A. The solution set is \( \{28, -2\} \).