Solve the linear inequality 3p − 16 < 20, and determine the correct range of values that the variable p can take.

Introduction / Context:
This question checks basic algebra skills involving linear inequalities. Instead of solving an equation with an equal sign, we solve an inequality that uses a less than relation. The method is similar to solving an equation, but we must keep track of the inequality sign when we add, subtract, multiply, or divide, especially when negative numbers are involved.

Given Data / Assumptions:

• The inequality given is 3p − 16 < 20.
• p is a real variable.
• No restriction is given that would require p to be an integer only.
• All operations are standard algebraic operations on real numbers.

Concept / Approach:
To solve a linear inequality, we isolate the variable on one side by using inverse operations, similar to solving linear equations. We can add or subtract the same number on both sides without changing the inequality direction. We can also divide by a positive number without changing the direction of the inequality. Here, all coefficients are positive, so the inequality sign will not reverse at any step.

Step-by-Step Solution:
Start with the inequality: 3p − 16 < 20.Add 16 to both sides to remove the constant term from the left: 3p − 16 + 16 < 20 + 16.This simplifies to 3p < 36.Divide both sides by 3, which is positive: p < 36 / 3.Hence p < 12.

Verification / Alternative check:
Pick a value less than 12, for example p = 10. Substitute: 3 * 10 − 16 = 30 − 16 = 14, and 14 < 20 is true. Pick a value equal to 12. For p = 12, 3 * 12 − 16 = 36 − 16 = 20, and 20 < 20 is false because it is equal, not less than. So p cannot be 12. For p greater than 12, for example 13, we get 3 * 13 − 16 = 39 − 16 = 23, which is not less than 20. This confirms that p must be less than 12.

Why Other Options Are Wrong:
Option a, p ≥ 12, contradicts our check because p = 12 does not satisfy the inequality. Option b, p ≤ 11/3, is too restrictive and does not include valid values like p = 10. Option d, p > 11/3, includes values greater than or equal to 12 which fail. Option e, p > 12, completely ignores values just below 12 that are still valid. Only p < 12 represents the correct solution set.

Common Pitfalls:
Students sometimes treat the inequality like an equation and include equality by mistake, writing p ≤ 12. Others may incorrectly divide by 3 and produce a wrong fraction. Remember that only when multiplying or dividing by a negative number must we reverse the inequality sign. Here that is not required, so the direction remains unchanged.

Final Answer:
The correct solution set is p < 12.