Final answer:
To solve the cubic inequality x³ + 3x² - 10x ≤ 19, we need to find the roots of the equation x³ + 3x² - 10x - 19 = 0 and then determine the intervals where the inequality holds. The solution can then be expressed in interval notation.
Step-by-step explanation:
To solve the inequality x³ + 3x² - 10x ≤ 19, we need to first bring the inequality to a standard form by subtracting 19 from both sides, resulting in x³ + 3x² - 10x - 19 ≤ 0. This is not a standard quadratic equation, so we cannot directly apply the quadratic formula. Instead, we look for real numbers that satisfy the equation x³ + 3x² - 10x - 19 = 0. Once we find such numbers, we can use them to determine the intervals where the inequality holds true.
We can use techniques such as factoring, graphing, or numerical methods to find the roots of the cubic equation. After determining the roots, we plot these on a number line and test intervals between the roots to see where the original inequality is satisfied. This process helps us find the solution in interval notation.
For example, if we find that the inequality is satisfied between two consecutive roots, such as between a and b, and that it doesn't hold for values less than a or greater than b, the interval notation would be: [a, b].