Problem 36 - Thomas Calculus

Question

Solve the inequalities in Exercises \(35-42 .\) Express the solution sets as intervals or unions of intervals and show them on the real line. Use the result \(\sqrt{a^{2}}=|a|\) as appropriate. $$ 4 \leq x^{2} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Inequality

The given inequality is \( 4 \leq x^2 \). We need to find all values of \( x \) such that \( x^2 \) is greater than or equal to 4. This involves both positive and negative roots because squaring a negative number gives a positive result.

Step 2: Use the Result \(\sqrt{a^2} = |a|\)

We know that \( \sqrt{x^2} = |x| \), which means \( |x| = \sqrt{4} \). This simplifies to \( |x| = 2 \). This implies that the solutions for \( x \) are when \( x = 2 \) or \( x = -2 \). However, as this is an inequality, we need to consider where \( |x| \geq 2 \).

Step 3: Solve the Absolute Value Inequality

From \( |x| \geq 2 \), we have two separate inequalities: \( x \geq 2 \) or \( x \leq -2 \). These inequalities will help us express the solution set for \( x \) in interval notation.

Step 4: Express Solution in Interval Notation

From the inequalities derived, we have \( x \geq 2 \) or \( x \leq -2 \). In interval notation, these solutions correspond to \( x \in (-\infty, -2] \cup [2, \infty) \).

Step 5: Visualize Solution on the Real Line

On the real line, the solution set \((-\infty, -2] \cup [2, \infty)\) consists of all numbers less than or equal to \(-2\) and all numbers greater than or equal to \(2\). Plot intervals using a closed dot at \(-2\) and another closed dot at \(2\), shading left for \( x \leq -2 \) and right for \( x \geq 2 \).

Key Concepts

Interval Notation

Interval notation is a way to describe a set of numbers that fall between two endpoints. These endpoints can be included or excluded from the set. In the inequality \(4 \leq x^2\), the goal is to find values for \(x\) such that \(x^2\) is greater than or equal to \(4\).

• The solution \(x \geq 2\) is expressed in interval notation as \([2, \infty)\), where \([\) indicates inclusion of \(2\), and \(\infty\) indicates the interval continues indefinitely.
• Similarly, \(x \leq -2\) is expressed as \((-\infty, -2]\), meaning all numbers less than or equal to \(-2\).

Interval notation provides a concise way of expressing the solution as \((-\infty, -2] \cup [2, \infty)\), indicating where \(x\) satisfies the inequality.

Real Line Representation

Representing numbers on the real line helps visually convey where solutions to an inequality reside. In this problem, we need to show the intervals where \(x\) satisfies the inequality \(4 \leq x^2\). To do this, we plot:

• A solid or closed dot at \(-2\), indicating \(x\) can be equal to \(-2\).
• A solid dot at \(2\) since \(x\) can also be equal to \(2\).
• Shading to the left of \(-2\) indicates all numbers less than or equal to \(-2\) are included in the solution.
• Similarly, shading to the right of \(2\) means all numbers greater than or equal to \(2\) are part of the solution.

Using a real line diagram is a helpful way to visualize how the solution includes two separate intervals on the number line.

Absolute Value Inequality

The key to solving inequalities like \(4 \leq x^2\) is understanding absolute value inequalities. By definition, if \(|x| \geq 2\), it implies that the magnitude or distance of \(x\) from zero is at least \(2\). To solve for \(x\), we break the absolute value inequality into two linear inequalities:

• When \(x \geq 2\), the values of \(x\) are greater than or equal to \(2\).
• When \(x \leq -2\), the values of \(x\) are less than or equal to \(-2\).

These inequalities together form the basis for finding intervals in the solution set. It's important to remember that solving the absolute value inequality helps separate the values of \(x\) into regions of the number line satisfying the original condition of the inequality.