Problem 39 - Calculus Single Variable

Question

Sketch the set on a real number line. \(\left\\{x:\left|x^{2}-5\right| \geq 4\right\\}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Inequality

The expression we are dealing with is \(|x^2 - 5| \geq 4\). This expression represents the set of values where the absolute value of \(x^2 - 5\) is greater than or equal to 4. This implies two possible conditions: \(x^2 - 5 \geq 4\) or \(x^2 - 5 \leq -4\).

Step 2: Solve the First Inequality

Let's solve \(x^2 - 5 \geq 4\). This simplifies to \(x^2 \geq 9\). Taking the square root of both sides, we get two solutions: \(x \geq 3\) and \(x \leq -3\).

Step 3: Solve the Second Inequality

Now solve \(x^2 - 5 \leq -4\). This simplifies to \(x^2 \leq 1\). Taking the square root of both sides, the solutions are \(-1 \leq x \leq 1\).

Step 4: Combine the Solutions

The solution to the absolute value inequality is the union of the two solutions from Steps 2 and 3. Therefore, the set is \((-\infty, -3] \cup [-1, 1] \cup [3, \infty)\).

Step 5: Sketch the Solution on a Number Line

To illustrate the solution on the number line, draw a bold line from \(-\infty\) to -3, a bold line from -1 to 1, and another bold line from 3 to \(+\infty\). Ensure closed circles at \(-3\), \(-1\), 1, and 3, indicating these values are included in the solution.

Key Concepts

Absolute Value Inequality

An absolute value inequality involves an expression wrapped in absolute value bars, such as \(|x^2 - 5| \geq 4\). Absolute value measures how far a number is from zero, regardless of direction. This means something like \(|a| \geq b\) translates to considering two cases: either the expression inside is greater than or equal to \(+b\) or less than or equal to \(-b\).
\[ \ |x^2 - 5| \geq 4 \text{ implies two scenarios: } \ x^2 - 5 \geq 4 \text{ or } x^2 - 5 \leq -4. \]
This double condition stems from the nature of absolute values encompassing both positive and negative possibilities. In our solved example, this resulted in two distinct expressions: \((x^2 - 5 \geq 4)\) simplifies to \((x^2 \geq 9)\), while \((x^2 - 5 \leq -4)\) simplifies to \((x^2 \leq 1)\). Understanding this dual-condition is crucial to mastering absolute value inequalities.

Quadratic Inequalities

Quadratic inequalities involve expressions with a degree of two, meaning the variable is squared (e.g., \(x^2 \geq 9\) or \(x^2 \leq 1\)).

Here's how we manage such inequalities:

• Identify opportunities to take the square root on both sides.


• This operation will yield two potential outcomes indicating ranges of solutions.


• For example, solving \(x^2 \geq 9\) gives \(x \geq 3\) and \(x \leq -3\).
• Conversely, for \(x^2 \leq 1\), we have \(-1 \leq x \leq 1\).

The difference between these two kinds of inequalities lies in whether they describe an interval (like \(-1 \leq x \leq 1)\) or two separate branches (\