Problem 45 - Algebra and Trigonometry

Question

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ x^{2}>3(x+6) $$

Step-by-Step Solution

Answer

Solve and graph the inequality, answer in interval notation.

Step 1: Set Up the Inequality

Rearrange the given inequality to standard form by moving all terms to one side. Start with the given inequality: \[ x^2 > 3(x + 6) \].First, distribute the 3 on the right-hand side:\[ x^2 > 3x + 18 \].Next, subtract \(3x + 18\) from both sides to get:\[ x^2 - 3x - 18 > 0 \].This inequality is in a form that can be analyzed using a sign test or graphing.

Step 2: Solve the Corresponding Equation

To determine the solution intervals, first solve the related quadratic equation:\[ x^2 - 3x - 18 = 0 \].This is solved using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a = 1\), \(b = -3\), and \(c = -18\).

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions where a quadratic polynomial is either greater than or less than a certain value or another polynomial. To solve them, you primarily examine the regions where the quadratic expression is positive or negative. A quadratic inequality such as \( x^2 - 3x - 18 > 0 \) requires similar steps as solving quadratic equations but focuses on finding the intervals where the inequality holds true.

The first step is to rearrange the inequality into a standard quadratic form, achievable by ensuring all terms are brought to one side. The expression is typically represented as \( f(x) > 0 \), \( f(x) < 0 \), or similar variations where \( f(x) \) is the quadratic polynomial. In our example, rearranging \( x^2 - 3(x + 6) > 0 \) gives us \( x^2 - 3x - 18 > 0 \).

Next, you solve the corresponding quadratic equation \( x^2 - 3x - 18 = 0 \) to find critical values that help outline the intervals for testing the inequality's truth. This helps to identify specific points that divide the number line into regions to check the sign of the expression.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill required to tackle quadratic inequalities. The process often involves factoring, using the quadratic formula, or completing the square. For the equation \( x^2 - 3x - 18 = 0 \), let's focus on employing the quadratic formula because it reliably works for all quadratic equations.

The quadratic formula is expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). It allows you to find the roots of any quadratic equation in the form \( ax^2 + bx + c = 0 \), by substituting the coefficients \( a \), \( b \), and \( c \) from the equation. Here, substituting \( a = 1 \), \( b = -3 \), and \( c = -18 \) into the formula results in:

• Calculate the discriminant: \( b^2 - 4ac = (-3)^2 - 4(1)(-18) = 81 \).
• The roots are then: \( x = \frac{3 \pm 9}{2} \), leading to \( x = 6 \) and \( x = -3 \).

These roots split the number line into intervals, which are essential for solving the quadratic inequality.

Interval Notation

Interval notation is a concise way of describing the solution set of inequalities, making it easy to understand the range of values that satisfy the condition. After identifying intervals from solving the quadratic equation, you test values within these intervals to determine where the inequality holds.

For instance, with roots at \( x = -3 \) and \( x = 6 \), the number line is divided into three regions: \((-\infty, -3)\), \((-3, 6)\), and \((6, \infty)\). By selecting test points from each interval, such as \( x = -4 \), \( x = 0 \), and \( x = 7 \), you can verify if the inequality \( x^2 - 3x - 18 > 0 \) holds true in those intervals. After testing, the inequality is satisfied in the intervals \((-\infty, -3)\) and \((6, \infty)\).

Using interval notation, you express this solution as \((-\infty, -3) \cup (6, \infty)\), clearly showcasing where the inequality holds. This system provides an efficient approach to writing solution sets, essential for clearly communicating which parts of the number line satisfy the original inequality.