Problem 39 - Algebra and Trigonometry

Question

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ x^{2}-3 x-18 \leq 0 $$

Step-by-Step Solution

Answer

The solution set is \([-3, 6]\) and can be graphed as a shaded region between \(-3\) and \(6\) on the number line.

Step 1: Identify the Quadratic Inequality

The given inequality is \(x^2 - 3x - 18 \leq 0\). We need to find the values of \(x\) that satisfy this inequality.

Step 2: Solve the Corresponding Quadratic Equation

First, solve the equation \(x^2 - 3x - 18 = 0\). This will help us identify the boundary points of the inequality. Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 1\), \(b = -3\), and \(c = -18\).

Step 3: Calculate the Roots Using the Quadratic Formula

Plug the values into the quadratic formula: \[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-18)}}{2 \cdot 1} \]Simplify inside the square root: \[ x = \frac{3 \pm \sqrt{9 + 72}}{2} \]\[ x = \frac{3 \pm \sqrt{81}}{2} \]\[ x = \frac{3 \pm 9}{2} \]Thus, the roots are \(x = 6\) and \(x = -3\).

Step 4: Test Intervals Between Roots

The roots divide the number line into intervals: \((-\infty, -3)\), \([-3, 6]\), and \((6, \infty)\). Test a point within each interval to see where the inequality holds. Use \(x = -5\), \(x = 0\), and \(x = 7\) for testing.

Step 5: Evaluate the Function in the Test Intervals

- For \(x = -5\), compute \((-5)^2 - 3(-5) - 18 = 25 + 15 - 18 = 22\) (not less than or equal to 0).- For \(x = 0\), compute \(0^2 - 3(0) - 18 = -18\) (less than 0).- For \(x = 7\), compute \(7^2 - 3(7) - 18 = 49 - 21 - 18 = 10\) (not less than or equal to 0).Only the interval \([-3, 6]\) satisfies the inequality.

Step 6: Write the Solution Set in Interval Notation

The solution set, where the inequality \(x^2 - 3x - 18 \leq 0\) holds, is \([-3, 6]\).

Step 7: Graph the Solution Set

On the number line, mark \(-3\) and \(6\) with solid dots to include these endpoints, and shade the region between them. This represents the interval \([-3, 6]\) where the inequality holds.

Key Concepts

Solutions to Inequalities

To solve a quadratic inequality, like \(x^2 - 3x - 18 \leq 0\), the process involves finding the values of \(x\) for which the inequality is true. Quadratic inequalities can result in an interval or several intervals where the inequality holds.
Steps typically include:

• Finding boundary points by solving the related equation \(x^2 - 3x - 18 = 0\).
• Dividing the number line into intervals based on these boundary points.
• Testing points from each interval in the inequality to see where it holds true.

This systematic method ensures all possible solutions are considered, making it easier to identify the correct interval where the inequality applies.

Interval Notation

Interval notation is a convenient way to express the solution set of an inequality. Once the appropriate intervals are identified, we use interval notation to present the solution clearly.
For the inequality \(x^2 - 3x - 18 \leq 0\), the solution is the interval

• \([-3, 6]\): This includes all values from -3 to 6, indicating both endpoints are part of the solution.

In interval notation:

• Square brackets, like \([\), indicate the endpoint is included (\(\leq\) or \(\geq\)).
• Parentheses, like \(()\), mean the endpoint is not included (\( < \) or \( > \)).

This compact form is widely used in mathematics, providing a neat summary of all solutions.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). The formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]To solve \(x^2 - 3x - 18 = 0\), we identify:

• \(a = 1\)
• \(b = -3\)
• \(c = -18\)

Substituting into the formula gives the two solutions, or roots, \(x = 6\) and \(x = -3\). These roots become the boundary points for our inequality, helping to divide the number line for further analysis. By using the quadratic formula, one can always find the necessary points to work further on solving the inequality.

Boundary Points

Boundary points are the solutions to the quadratic equation derived from the inequality. They indicate points on the number line where the behavior of the quadratic function changes.
For \(x^2 - 3x - 18 \leq 0\), solving \(x^2 - 3x - 18 = 0\) yields boundary points \(x = -3\) and \(x = 6\). The significance of boundary points includes:

• They help divide the number line into testable intervals.
• They can be endpoints of intervals in our solution set if included by the inequality (such as \(\leq\) or \(\geq\)).

These points mark key positions and guide through which sections of a graph might satisfy the inequality.

Number Line Graphing

Graphing the solution on a number line gives a visual representation of the intervals where the inequality is satisfied. For \(x^2 - 3x - 18 \leq 0\), you would:

• Mark the boundary points \(-3\) and \(6\) with solid dots because the inequality includes these points (\(\leq\)).
• Shade the portion of the number line between these points to illustrate the interval \([-3, 6]\).

Number line graphing is a straightforward way to present solutions and easily verify which intervals are part of the solution set. It provides an intuitive view of how the quadratic expression behaves over different values.