Problem 26 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality. Then graph the solution set on the real number line. \(x^{3}+5 x^{2}-4 x-20 \leq 0\)

Step-by-Step Solution

Answer

The solution of the inequality is \(x \in (-\infty, -5.1926] \cup (1.3342, 6.8584]\). The graph would show closed circles at -5.1926 and 6.8584 with the sections between -5.1926 and 1.3342 (excluding 1.3342) and above 6.8584 shaded to indicate where the inequality holds true.

Step 1: Simplify the inequality

The inequality given is a cubic polynomial inequality. The polynomial is already simplified, so there’s no need to simplify it further.

Step 2: Find the roots

To find the roots of the equality \(x^{3}+5x^{2}-4x-20=0\), try to factorize it. Unfortunately, this polynomial cannot be factorized into simpler polynomials with rational coefficients. Therefore, we need to use third degree polynomial solution formula or software to approximate the roots numerically: approximately -5.1926, 1.3342, and 6.8584.

Step 3: Test the intervals

After the roots have been found, place them on a number line. This separates the number line into intervals. Test each interval with a test point to see if the inequality holds true. Based on the number line, the intervals are \(-\infty,-5.1926\), \(-5.1926, 1.3342\), \(1.3342, 6.8584\), and \(6.8584, +\infty\). Choose test points from each interval, say, \(-6, 0, 4, 7\) respectively and plug them into the inequality. If it holds true, the inequality is true for all numbers in that interval.

Step 4: Write the solution

Based on interval testing, the solution to the inequality should be in interval notation which intervals makes the inequality holds true.

Step 5: Graph the inequalities

Plot the solution intervals on a number line. Put an open circle on a number if it's not included (greater/less than) and a closed circle if it's included (greater/less than or equal to).

Key Concepts

Solving Inequalities

Solving inequalities involves finding the values of a variable that make the inequality true. A cubic polynomial inequality like \(x^{3}+5x^{2}-4x-20 \<= 0\) is more complex than linear or quadratic inequalities but follows similar principles.

To solve, the first step is often to identify the roots of the polynomial, which are the values of \(x\) where the polynomial equals zero. These roots serve as critical points that divide the number line into intervals - each of which must be tested to determine whether the inequality is satisfied. The solution to the inequality will be a range or ranges of values where the inequality holds true. For a less than or equal to (\(\leq\)) inequality, the values at the roots are included in the solution set.

Graphing on Real Number Line

Graphing on the real number line helps visualize the solution to inequalities. To illustrate the solutions to the inequality \(x^{3}+5x^{2}-4x-20 \<= 0\), the real number line is divided based on the roots or critical points. Each interval between and beyond the roots is analyzed to determine where the inequality is true.

When graphing, open or closed circles represent whether a root is part of the solution. An open circle indicates that the value at that point is not included, while a closed circle means it is included. The solution set is then shown as a continuous line or ray over the intervals that satisfy the inequality.

Roots of Polynomial

Finding the roots of a polynomial is crucial for solving polynomial inequalities. Roots are the solutions to the equation when the polynomial is set equal to zero. In the given exercise \(x^{3}+5x^{2}-4x-20 = 0\), the roots are the x-values where the polynomial intersects the x-axis on a graph.

For cubics that are not easily factorizable, numerical methods or graphing calculators are often used to approximate the roots. Knowledge of these roots allows us to divide the number line into meaningful intervals for further analysis when solving inequalities.

Interval Notation

Interval notation is an efficient way to describe the set of solutions to an inequality. This notation uses parentheses and brackets to convey which numbers are included in the set. A bracket \([\) or \(]\) indicates the endpoint is included, equivalent to 'less than or equal to' (\(\leq\)) or 'greater than or equal to' (\(\geq\)), while a parenthesis \((\) or \()\) shows the endpoint is not included.

For instance, the interval \(-\infty, a)\) represents all numbers less than \(a\), but not including \(a\) itself. Combining our solutions into an interval, we express continuous ranges where the inequality is true, ensuring a concise and clear representation of the solution.