Problem 33 - Algebra and Trigonometry

Question

\(33-66\) me solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ (x+2)(x-3)<0 $$

Step-by-Step Solution

Answer

The solution is the interval \((-2, 3)\).

Step 1: Identify Critical Points

To solve the inequality \((x+2)(x-3)<0\), first identify the points where each factor equals zero. These points, known as critical points, are found by setting \(x+2=0\) and \(x-3=0\). Solving these equations gives:\[x+2=0 \Rightarrow x=-2\] and \[x-3=0 \Rightarrow x=3\]. These critical points divide the number line into intervals.

Step 2: Test Intervals

The critical points \(-2\) and \(3\) divide the number line into three intervals: \((-\infty, -2)\), \((-2, 3)\), and \((3, \infty)\). We need to test a point within each interval to determine where the inequality \((x+2)(x-3)<0\) holds.1. Choose \(x=-3\) for \((-\infty, -2)\): \((-3+2)(-3-3) = (-1)(-6) = 6 > 0\)2. Choose \(x=0\) for \((-2, 3)\): \((0+2)(0-3) = 2(-3) = -6 < 0\)3. Choose \(x=4\) for \((3, \infty)\): \((4+2)(4-3) = 6(1) = 6 > 0\)The inequality is satisfied in the interval \((-2, 3)\).

Step 3: Write Solution in Interval Notation

Since the inequality holds in the interval \((-2, 3)\), we write the solution in interval notation considering that \(-2\) and \(3\) are not included in the solution because the inequality is strict (\(<\) and not \(\leq\)). Thus, the solution is \((-2, 3)\).

Step 4: Graph the Solution Set

Graph the solution set \((-2, 3)\) on a number line by drawing an open circle at \(-2\) and \(3\) to denote that these points are not included, and shading the region between them to show that \(x\) values within this interval satisfy the inequality.

Key Concepts

Critical Points

In the context of solving inequalities, identifying critical points is crucial. These points exist where the factors of an expression equal zero. We need these points to analyze how the inequality behaves around them.
For the inequality \((x+2)(x-3)<0\), the critical points occur when each factor equals zero:

• Setting \(x+2=0\), we find \(x=-2\).
• Setting \(x-3=0\), we find \(x=3\).

These critical points, \(-2\) and \(3\), divide the number line into distinct intervals. Understanding where these points lie is key to identifying where the inequality holds true. Without locating these critical points accurately, it would be difficult to solve or interpret the inequality solution effectively.

Interval Notation

Interval notation is a shorthand way of writing subsets of the real number line, making it particularly useful in expressing the solution sets of inequalities.
When we talk about solving an inequality such as \((x+2)(x-3)<0\), interval notation helps us communicate the range of \(x\) values that satisfy the inequality.
For example, after finding that the inequality holds true between the critical points \(-2\) and \(3\), we express this range as \((-2, 3)\):

• The parentheses \(( \) and \() \) indicate that the endpoints \(-2\) and \(3\) are not included because the inequality is strict.

This notation effectively conveys that the solution includes every number between \(-2\) and \(3\), excluding \(-2\) and \(3\) themselves. Understanding interval notation is vital as it offers clarity and precision in conveying solution sets.

Testing Intervals

Once we identify the critical points, \(-2\) and \(3\), the number line is divided into three intervals:

• \((-\infty, -2)\)
• \((-2, 3)\)
• \((3, \infty)\)

To determine where the inequality \((x+2)(x-3)<0\) holds, we test points within each interval.
Testing involves substituting a value from each interval into the inequality:

• For \((-\infty, -2)\), using \(x=-3\) results in a positive value, indicating the inequality does not hold there.
• For \((-2, 3)\), with \(x=0\), the result is negative, showing this interval satisfies the inequality.
• For \((3, \infty)\), testing \(x=4\) gives a positive value, thus not holding for the inequality.

This systematic approach ensures we identify the precise intervals where the inequality is true.

Graphing Inequalities

Graphing inequalities provides a visual representation of the solution set. This graphing process involves marking the solution interval on a number line.
For \((x+2)(x-3)<0\), we've determined the solution to be \((-2, 3)\). This can be depicted by:

• Drawing open circles at points \(-2\) and \(3\) on the number line, indicating these boundaries are not part of the solution.
• Shading the interval between \(-2\) and \(3\) to show that all numbers in this range satisfy the inequality.

This graphical representation aligns with the interval notation \((-2, 3)\), helping us comprehend the solutions more intuitively. Understanding how to graph inequalities is beneficial, reinforcing the connection between algebraic solutions and their visual forms.