Problem 45 - Precalculus Mathematics for Calculus

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

The solution in interval notation is \((-\infty, -3) \cup (6, \infty)\).

Step 1: Simplify the Inequality

The first step involves expanding and rearranging the inequality. Start with the given inequality: \[ x^2 > 3(x + 6) \].Expand the right-hand side:\[ x^2 > 3x + 18 \].Subtract \(3x\) and \(18\) from both sides to set the inequality to zero:\[ x^2 - 3x - 18 > 0 \].

Step 2: Find the Critical Points

Next, solve the equation \[ x^2 - 3x - 18 = 0 \] to find the critical points.Use the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\], where \(a = 1\), \(b = -3\), and \(c = -18\).Calculate the discriminant:\[b^2 - 4ac = (-3)^2 - 4(1)(-18) = 9 + 72 = 81\].Find the roots:\[x = \frac{3 \pm \sqrt{81}}{2} = \frac{3 \pm 9}{2}\].This gives us two roots: \(x_1 = 6\) and \(x_2 = -3\).

Key Concepts

Interval Notation

Interval notation is a way of expressing the set of solutions to inequalities. It uses intervals to show which parts of the number line are included in the solution. This is especially useful for understanding nonlinear inequalities, where solutions are not limited to single points.

Here's how interval notation works:

• Use parentheses, \(( \text{ or } )\), to describe an interval that does not include the endpoints, also known as open intervals.
• Use brackets, \([ \text{ or } ]\), to define an interval that includes the endpoints, which are closed intervals.
• Combine these symbols to demonstrate solutions that span multiple intervals.

For example, in our original exercise, the inequality \(x^2 > 3x + 18\) translates into finding intervals where the expression \(x^2 - 3x - 18 > 0\). The solution is not a continuous line but two separate intervals: \((-\infty, -3) \cup (6, \infty)\). This shows that any \(x\) value less than \(-3\) or greater than \(6\) satisfies the inequality, expressed using interval notation.

Graph Solution Set

When solving inequalities, it’s often helpful to visualize the solution on a graph, which is known as graphing the solution set. This allows you to see on the number line where the solutions to the inequality lie, forming a clear picture of which intervals or values are included.

To graph the solution set:

• Identify critical points (found from the equation \(x^2 - 3x - 18 = 0\)). These points are \(x = -3\) and \(x = 6\) in this problem.
• Include these critical points as markers on the number line.
• Check the intervals between and outside these points. Remember that the inequality \(x^2 - 3x - 18 > 0\) suggests testing regions: \((-\infty, -3)\), \((-3, 6)\), and \((6, \infty)\).
• Solve these sections to determine where the inequality holds true. Shade the regions where the values satisfy the inequality. In this exercise, the intervals \((-\infty, -3)\) and \((6, \infty)\) are shaded.

The graph provides a visual representation that complements the interval notation, making it easier to see the range of solutions at a glance.

Quadratic Formula

The quadratic formula is a tool for finding the solutions, or roots, of quadratic equations. A quadratic equation is generally expressed as \(ax^2 + bx + c = 0\). When equations become complex or factoring is infeasible, the quadratic formula is a reliable method.

The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here:

• \(a\), \(b\), and \(c\) are coefficients from the quadratic equation.
• \(\pm\) indicates there can be two possible values for \(x\).
• The term \(b^2 - 4ac\) is known as the discriminant, determining the number and nature of roots.

In our exercise, the equation \(x^2 - 3x - 18 = 0\) means using the quadratic formula requires:\(a = 1\), \(b = -3\), and \(c = -18\). Calculating the discriminant, we find it is 81, indicating two real and distinct roots.

Simplifying using the formula gives the roots \(x_1 = 6\) and \(x_2 = -3\), which are essential in determining the critical points for the inequality \(x^2 - 3x - 18 > 0\). These roots help divide the number line into intervals, guiding the solution set for the inequality.