Problem 47 - Precalculus Mathematics for Calculus

Question

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$x^{2}<4$$

Step-by-Step Solution

Answer

The solution is \((-2, 2)\). Graphically, shade the region between -2 and 2 with open circles at the endpoints.

Step 1: Understand the inequality

We start with the inequality \( x^2 < 4 \), which represents the set of numbers \( x \) whose square is less than 4. Our goal is to find the values of \( x \) that satisfy this condition.

Step 2: Solve for x

To solve \( x^2 < 4 \), first recognize that \( x^2 = 4 \) gives \( x = \pm 2 \). Thus, the inequality becomes: \(-2 < x < 2\). This is because the square root of 4 is 2, and we need \( x^2 \) to be less than 4.

Step 3: Write the solution in interval notation

The solution to the inequality is the interval of \( x \) values that lie between -2 and 2, but do not include -2 and 2 themselves. Therefore, we express the solution in interval notation as \((-2, 2)\).

Step 4: Graph the solution set

To graph the solution \((-2, 2)\), draw a number line, and shade the region between -2 and 2. Use open circles at -2 and 2 to indicate that these points are not included in the solution set.

Key Concepts

Understanding Nonlinear Inequalities

Nonlinear inequalities involve expressions that are more complex than simple linear relationships. In our case, we have the inequality \( x^2 < 4 \). This inequality is nonlinear because it includes a squared term. Understanding this inequality means recognizing it describes a set of \( x \) values whose squares produce results less than 4. The key difference from linear inequalities is that instead of forming a straight line, the solution creates a parabolic area on a graph.
Nonlinear inequalities require specific techniques to solve because they do not produce a straight line solution. In our example, we first determine the bounds by solving the equation \( x^2 = 4 \), which provides critical points \( x = -2 \) and \( x = 2 \). These points define the boundary of the region where \( x^2 \) is less than 4.

Using Interval Notation

Interval notation is a helpful tool to succinctly express the range of values that satisfy an inequality. It uses brackets and parentheses to show whether endpoints are included or excluded.In our inequality \( x^2 < 4 \), the solution concerns values of \( x \) strictly between -2 and 2. Therefore, we use parentheses to indicate that the endpoints -2 and 2 are not included in our solution, expressed as \((-2, 2)\).

• Parentheses \((\cdot)\) are used for open intervals, meaning the endpoints themselves are not part of the solution.
• Brackets \([\cdot]\) would indicate closed intervals, where the endpoints are included.

Interval notation effectively communicates the solution, helping us quickly determine which \( x \) values satisfy the nonlinear inequality.

Graphing the Solutions

Graphing the solutions of an inequality provides a visual representation of the solution set. For the inequality \( x^2 < 4 \), we graph this on a number line.Firstly, plot the critical points, \(-2\) and \(2\), with open circles. Open circles indicate that these points, while boundary markers, are not included in the solution set.
The next step involves shading the area between these open circles. The shaded region shows all possible values of \( x \) that, when squared, remain less than 4. Graphing makes it easier for students to understand which numbers are solutions. It also visually confirms the interval notation \((-2, 2)\). This method not only reinforces the solution but also provides an insightful check that prevents misunderstanding of purely symbolic answers.