Problem 28 - Calculus: One and Several Variables

Question

Indicate on a number line the numbers \(x\) that satisfy the condition. \(x^{2} \geq 16\).

Step-by-Step Solution

Answer

The solution to the given inequality \(x^{2} \geq 16\) is \(x \geq 4\) or \(x \leq -4\). On a number line, this can be represented with arrows pointing towards right from 4 and towards left from -4 with solid circles at these points.

Step 1: Solve the Quadratic Inequality

We first solve the inequality \(x^{2} \geq 16\). We take the square root of both sides, remembering that when we remove the square on the left, it becomes \(|x|\), so we get \(|x| \geq 4\). The absolute value inequality breaks into two parts: \(x \geq 4\) and \(x \leq -4\).

Step 2: Solution and Its Implication

The solutions \(x \geq 4\) and \(x \leq -4\) implies that all real numbers greater than or equal to 4 or less than or equal to -4 are valid solutions for the given inequality.

Step 3: Plotting on the Number Line

We plot these solutions on a number line indicating with arrows that all values greater than or equal to 4 and less than or equal to -4 are solutions. Arrow towards right from 4 will represent \(x \geq 4\) and arrow towards left from -4 will represent \(x \leq -4\). Place a solid circle on 4 and -4 to indicate that these numbers are also included as solutions.

Key Concepts

Understanding Absolute Value Inequalities

When we talk about absolute value inequalities, we're dealing with expressions that involve the distance between a number and zero on the number line. Absolute value is represented by vertical bars like this: \(|x|\). For the inequality \(|x| \geq 4\), we're looking for all numbers whose distance from zero is at least 4. This means the numbers can either be greater than or equal to 4 or less than or equal to -4. In absolute value inequalities, the inequality can be split into two separate inequalities that address both the positive and negative solutions. This particular inequality splits into \(x \geq 4\) and \(x \leq -4\). This dual condition arises because absolute value represents distance, which is always non-negative and can have solutions on both sides of zero.

To solve absolute value inequalities:

• Identify the expression inside the absolute value.
• Set up two separate inequalities based on the original absolute value relation: one for the positive scenario and one for the negative.
• Solve both inequalities to find the set of solutions.

Finding Inequality Solutions

Solving inequalities isn't too different from solving equations, but with an important twist. Inequality solutions require understanding how to handle inequality signs, especially when dealing with negative numbers or multiplication/division. Let's take \(x^{2} \geq 16\) as an example. First, recognize that the left side can be rewritten as \(|x| \geq 4\) by taking the square root of both sides. This shifts the problem to solving an absolute value inequality.

So, break this into two clear inequalities:

• \(x \geq 4\)
• \(x \leq -4\)

Solving inequalities means identifying all x-values that make the inequality true. In the case of \(x^{2} \geq 16\), we find that the solutions include any \(x\) that is less than or equal to -4 or greater than or equal to 4. These solutions reflect the idea of distance in absolute value and highlight the importance of considering both ends of the number line.

Visualizing with Number Line Representation

The number line is a fantastic tool for visualizing inequalities and their solutions. It helps us see the range of numbers that satisfy a given condition such as \(x^{2} \geq 16\). By drawing a number line, we can represent solutions graphically, which often clarifies complex concepts.

Here's how to represent the solution of \(x^{2} \geq 16\) on a number line:

• Begin by marking the critical points on the line, which in this case are 4 and -4.
• Draw a solid circle on these points to show that they are included in the solution (because our inequality is ">=", meaning inclusive).
• From 4, draw an arrow extending to the right to indicate that all numbers greater than or equal to 4 satisfy the inequality.
• From -4, draw an arrow extending to the left to indicate all numbers less than or equal to -4 are solutions.

This clear visual representation on the number line helps reinforce the understanding that the solutions cover two distinct intervals on the real number line, capturing exactly how these numbers are distributed and related to the original quadratic inequality.