Problem 49 - Algebra and Trigonometry

Question

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ (x+2)(x-1)(x-3) \leq 0 $$

Step-by-Step Solution

Answer

The solution is \([-2, 1] \cup [1, 3]\).

Step 1: Identify Zero Points

Find the points where each factor of the expression equals zero. For the inequality \((x+2)(x-1)(x-3) \leq 0\), set each factor to zero: 1. \(x + 2 = 0\) gives \(x = -2\).2. \(x - 1 = 0\) gives \(x = 1\).3. \(x - 3 = 0\) gives \(x = 3\).These points divide the number line into intervals.

Step 2: Test Intervals

Based on the zero points \(x = -2, 1, 3\), we divide the number line into the intervals \((-\infty, -2), (-2, 1), (1, 3), (3, \infty)\). Test a point from each interval in the inequality \((x+2)(x-1)(x-3)\). We are looking for where the expression is less than or equal to zero.1. Choose \(x = -3\) for \((-\infty, -2)\): \((-1)(-4)(-6) > 0\)2. Choose \(x = 0\) for \((-2, 1)\): \((2)(-1)(-3) < 0\)3. Choose \(x = 2\) for \((1, 3)\): \((4)(1)(-1) < 0\)4. Choose \(x = 4\) for \((3, \infty)\): \((6)(3)(1) > 0\)The expression is less than or equal to zero for the intervals \((-2, 1)\) and \((1, 3)\).

Step 3: Include Zero Points

Since the problem is \(\leq 0\) (including zero), we also consider the zero values at the detected points where the expression changes sign. The expression equals zero at \(x = -2, 1, 3\). Therefore, these points are included in the solution.

Step 4: Write the Solution in Interval Notation

Combine the intervals where the inequality holds: The inequality is less than zero on \((-2, 1)\) and \((1, 3)\), and exactly zero at \(x = -2, 1,\) and \(3\). Hence, the complete solution in interval notation is \([-2, 1]\cup[1, 3]\).

Step 5: Graph the Solution Set

Draw a number line and mark the points \(x = -2, 1, 3\) with solid circles or squares, as these points are included in the solution. Shade the intervals \([-2, 1]\) and \([1, 3]\) to represent the solution set graphically.

Key Concepts

Zero Points

In any inequality, especially nonlinear ones, identifying the zero points is one of the first crucial steps. These zero points indicate where the expression equals zero. For our specific inequality \((x+2)(x-1)(x-3) \leq 0\), we find these zero points by setting each factor equal to zero:

• \(x + 2 = 0\) gives \(x = -2\)
• \(x - 1 = 0\) gives \(x = 1\)
• \(x - 3 = 0\) gives \(x = 3\)

These solutions partition the number line into segments that will be used for further testing. The points \(-2\), \(1\), and \(3\) can also be described as critical values, where the nature of the inequality might change. Understanding and finding zero points help us know where the product of the factors potentially changes sign. This foundational step is crucial for identifying valid intervals of the inequality.

Interval Notation

Interval notation is a succinct way to describe a set of numbers, especially useful in inequalities. After identifying valid intervals where the inequality holds true or equals zero, we express the solution using this notation. For our inequality example, we determine the expression is less than or equal to zero in the intervals:

• \((-2, 1)\)
• \((1, 3)\)

Additionally, because our inequality is \(\leq 0\), we include the zero points, which means these intervals are closed at \(-2, 1,\) and \(3\). The combined interval notation, including these endpoints, is written as \([-2, 1] \cup [1, 3]\). Here, square brackets \([]\) indicate the number is included in the solution set, while parentheses \(()\) would signify excluding a number. Using interval notation helps communicate solutions clearly and concisely, avoiding potential ambiguity.

Graphical Solution

A graphical solution involves visually representing the set of solutions on a graph or number line. This gives a clear view of where the inequality holds. For the inequality \((x+2)(x-1)(x-3) \leq 0\), we have determined that the solution set includes the intervals \([-2, 1]\) and \([1, 3]\). To create a graphical solution:

• Mark the zero points \(x = -2\), \(x = 1\), and \(x = 3\) on a number line with solid circles. This indicates they are included due to the \(\leq 0\) aspect of the inequality.
• Shade the intervals between and including these points: \([-2, 1]\) and \([1, 3]\). This shading shows that any \(x\) value within these intervals is a valid solution to the inequality.

The graphical solution is not only intuitive but powerful, offering an immediate understanding of solution intervals. It helps in quickly determining possible candidates for \(x\) that meet inequality requirements.

Number Line

The number line is a fundamental mathematical tool for exploring solutions to inequalities. It allows for easy visualization and evaluation of different regions created by zero points.In solving \((x+2)(x-1)(x-3) \leq 0\), the number line is divided at these critical values \(-2\), \(1\), and \(3\). When using the number line:

• Place these points on the line to split it into segments \((-\infty, -2)\), \((-2, 1)\), \((1, 3)\), and \((3, \infty)\).
• Next, test each segment with a point to decide if the interval satisfies the inequality (consider whether the sign of the expression becomes negative or zero).
• The segments that make the inequality true are then marked and considered part of the solution set.

A number line transforms the abstract inequality into a tangible, structured visual guide. It's a simple but effective tool for analyzing how equations behave over different intervals of \(x\).