Final answer:
To solve the inequality (x-2)/(x²-16)<0, we identify the critical points and then test intervals between these points. The solution set consists of the intervals where the inequality is satisfied, which are (-4, 2) and (4, ∞), excluding the points where the denominator is zero.
Step-by-step explanation:
To solve the rational inequality (x-2)/(x²-16)<0, we need to find where the expression is negative. The denominator can be factored as (x+4)(x-4), so the critical points where the expression could change signs are at x = 2, x = -4, and x = 4. Now we look at the intervals created by these points to determine where the inequality holds true:
- Interval (-∞, -4): Choose a test value like x = -5. Plugging it into the inequality, we find the expression is positive, so this interval is not part of the solution.
- Interval (-4, 2): Choose a test value like x = 0. We then get a negative result, which tells us the expression is negative in this interval, thus part of our solution.
- Interval (2, 4): Test a value like x = 3. Plugging it in, we get a positive value, hence it's not part of the solution.
- Interval (4, ∞): Test with x = 5. The expression is negative again, so this interval is included in our solution.
Don't forget to consider the excluded values where the denominator equals zero (x = -4 and x = 4). These cannot be part of the solution set as they would make the original fraction undefined.
Therefore, our solution is the union of the intervals where the expression is negative: (-4, 2) U (4, ∞).