Final answer:
To solve the quadratic inequality algebraically using test regions, we need to rewrite the inequality in standard form, factorize it, determine the critical points, and use the test regions method to identify the solution.
Step-by-step explanation:
To solve the quadratic inequality algebraically using test regions, we need to first rewrite the inequality in the standard form ax^2 + bx + c < 0, where 'a' is positive.
In this case, the inequality is x^2 - 5x < 24. Subtracting 24 from both sides, we get x^2 - 5x - 24 < 0.
Next, we factorize the quadratic equation: (x - 8)(x + 3) < 0.
From the factors, we can determine the critical points where the inequality changes sign. So, x = -3 and x = 8.
Now, we can use the test regions method by choosing test points from each interval. When we plug in a value less than -3, like -4, into the original inequality, we get (-4 - 8)(-4 + 3) < 0, which gives us a positive value. When we plug in a value between -3 and 8, like 0, we get (0 - 8)(0 + 3) < 0, which gives us a negative value. When we plug in a value greater than 8, like 9, we get (9 - 8)(9 + 3) < 0, which gives us a positive value.
Therefore, the solution to the inequality is x ∈ (-∞, -3) ∪ (8, ∞).