Problem 39 - Precalculus Mathematics for Calculus

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 is \([-3, 6]\).

Step 1: Set the inequality as an equation

First, we need to solve the equation \(x^2 - 3x - 18 = 0\) to find the critical points. These points will help us determine where the inequality might change.

Step 2: Use the quadratic formula

The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). For the equation \(x^2 - 3x - 18 = 0\), we have \(a = 1\), \(b = -3\), and \(c = -18\). Substitute these values into the formula to find the roots.

Step 3: Calculate the discriminant

Calculate the discriminant \(b^2 - 4ac\). Substitute \(b = -3\), \(a = 1\), and \(c = -18\) into the expression: \((-3)^2 - 4(1)(-18) = 9 + 72 = 81\). The discriminant is 81, which is a perfect square, indicating we have two distinct roots.

Step 4: Solve for the roots

Since the discriminant is 81, apply the quadratic formula: \(x = \frac{3 \pm \sqrt{81}}{2}\). This gives us \(x = \frac{3 \pm 9}{2}\). So, the roots are \(x = 6\) and \(x = -3\).

Step 5: Determine the test intervals

The critical points \(x = -3\) and \(x = 6\) divide the number line into three intervals: \(( -\infty, -3 )\), \([-3, 6]\), and \((6, \infty)\). Choose a test point from each interval to determine where the inequality holds.

Step 6: Test the intervals

Select test points: \(x = -4\) for \((-\infty, -3)\), \(x = 0\) for \([-3, 6]\), and \(x = 7\) for \((6, \infty)\). Substitute these into \(x^2 - 3x - 18\) to check the sign in each interval.

Step 7: Analyze the sign of the polynomial

For \(x = -4\), \((-4)^2 - 3(-4) - 18 = 16 + 12 - 18 = 10\), which is positive. For \(x = 0\), \(0^2 - 3(0) - 18 = -18\), which is negative. For \(x = 7\), \(7^2 - 3(7) - 18 = 49 - 21 - 18 = 10\), which is positive.

Step 8: Write the solution in interval notation

The inequality \(x^2 - 3x - 18 \leq 0\) holds for the interval where the polynomial is zero or negative, which is \([-3, 6]\). This is the solution in interval notation.

Step 9: Graph the solution set

On a number line, mark the endpoints \(-3\) and \(6\) with brackets to include them in the solution. Shade the region between \(-3\) and \(6\). This shows where \(x^2 - 3x - 18 \leq 0\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form \[ ax^2 + bx + c = 0 \]. It is especially handy when the equation cannot be easily factored.

The formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Here, the term \( b^2 - 4ac \) is known as the discriminant.

This discriminant plays a crucial role:

• If it is positive, the equation has two distinct real roots.
• A discriminant of zero indicates exactly one real root.
• If negative, the equation has no real roots, instead, producing complex numbers.

In the exercise, after substituting values \( a = 1 \), \( b = -3 \), and \( c = -18 \), the discriminant becomes 81, which led to two real solutions: \( x = 6 \) and \( x = -3 \). The quadratic formula provides a systematic way to obtain these roots and identify critical points essential for analyzing the inequality's behavior.

Interval Notation

Interval notation is a concise way of representing a range of values, particularly useful for solution sets of inequalities.

There are different symbols to indicate whether endpoints are included or not:

• Brackets \([ ]\) show that endpoints are part of the interval (inclusive).
• Parentheses \(( )\) indicate that endpoints are not included (exclusive).

In the context of inequalities, using test intervals helps us determine the sign of a polynomial expression within different ranges divided by critical points. For the inequality \( x^2 - 3x - 18 \leq 0 \), testing led to finding the interval \([-3, 6]\), meaning that these values satisfy the inequality condition.

The solution in interval notation, thereby, becomes \([-3, 6]\), effectively combining the inclusive nature of all values between and including the roots \(x = -3\) and \(x = 6\).

Graphing Solution Sets

Graphing solution sets is a visual approach to understanding the range of solutions for a given inequality.

When we graph \( x^2 - 3x - 18 \leq 0 \), we start by marking the critical points identified earlier, which are the roots \( x = -3 \) and \( x = 6 \). On a number line:

• Use brackets at \(-3\) and \(6\) to include these endpoints.
• Shade the region between these points.

This shaded section represents all real numbers that solve the inequality within the given range.

Visualizing how the graph correlates to the inequality can provide deeper insight, making it easier to comprehend where solutions lie and how they interact with the function's behavior across the number line.