Problem 10 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality. Then graph the solution set on the real number line. \((x-3)^{2} \geq 1\)

Step-by-Step Solution

Answer

The solution set for the inequality \((x - 3)^{2} \geq 1\) equals to \((- \infty, 2] \cup [4, \infty)\).

Step 1: Remove the square root

To get started, remove the square root from \((x - 3)^2\). Thus, the inequality becomes \(x - 3 \geq \sqrt{1}\) and \(x - 3 \leq -\sqrt{1}\). That simplifies to \(x \geq 3 + 1 \) and \(x \leq 3 - 1 \).

Step 2: Solve for x

Solving the inequalities provides \(x \geq 4\) and \(x \leq 2\).

Step 3: Establish and test intervals

After identifying the roots, the next step is to establish the intervals using the identified roots and then test these intervals. The intervals established from this inequality are \((- \infty, 2]\), \((2, 4)\), and \([4, \infty)\). Choose any number within each interval and substitute it into the original inequality to test the intervals. For instance, you can pick \(1, 3,\) and \(5\) to represent the respective intervals. When \(x = 1\), \((1 - 3)^{2} = 4 \geq 1\) is true. When \(x = 3\), \((3 - 3)^{2} = 0 < 1\) is false. When \(x = 5\), \((5 - 3)^{2} = 4 \geq 1\) is true.

Step 4: Graph the solution set

Lastly, graph the solution set on a real number line. The graph will include the intervals \((- \infty, 2]\) and \([4, \infty)\).

Key Concepts

Quadratic Inequalities

When dealing with quadratic inequalities, such as \((x-3)^2 \geq 1\), you are working with a mathematical expression that involves an inequality sign together with a quadratic expression. Quadratic inequalities are similar to quadratic equations, but instead of an equal sign, they use inequality symbols like \(>\), \(<\), \(\geq\), or \(\leq\). The goal is to determine values of \(x\) that satisfy the inequality.

These expressions can represent a range of values rather than a specific set of points. A common approach to solving them is to first rewrite the inequality without the square terms by factoring or taking square roots on both sides. After simplifying, you solve for \(x\) to find the critical values.

With quadratic inequalities, it's important to consider the intervals between critical points to see where the inequality holds true.

Solution Set

The solution set of an inequality is the collection of all values that make the inequality true. For the inequality \((x-3)^2 \geq 1\), the solution set is the collection of \(x\) values that satisfy this inequality.

After removing the square root from the quadratic term and simplifying, the inequality splits into \(x \geq 4\) and \(x \leq 2\). This means the solution set includes all \(x\) values less than or equal to 2 and those greater than or equal to 4. Hence, the solution set can be expressed as the union of two intervals: \((−∞, 2]\) and \([4, ∞)\).

The solution set is not continuous, highlighting how quadratic inequalities often have disjointed regions of solution.

Real Number Line

The real number line is a visual representation that helps us understand the intervals where the solutions of an inequality lie. It is a line that contains all real numbers, extending infinitely in both directions.

For graphing inequalities like \((x-3)^2 \geq 1\), it's useful to show which parts of the number line are included in the solution set. On this line, you plot specific points that represent the critical values found in your solution. In this case, you’ll plot \(2\) and \(4\).

Next, darken the portions of the line corresponding to the intervals \((-\infty, 2]\) and \([4, \infty)\), indicating that these are part of the solution set.

Interval Testing

Interval testing is a technique used to determine which parts of the real number line satisfy the inequality. Once the critical points are known, the real number line is divided into distinct intervals.

For our inequality \((x-3)^2 \geq 1\), the critical points divide the real number line into the intervals \((-\infty, 2]\), \((2, 4)\), and \([4, \infty)\).

To test these intervals, select a sample point from each interval, substitute it back into the original inequality, and check if the inequality holds true. If it does, that interval is part of the solution set. For example, testing \(x = 1\) for the interval \((-\infty, 2]\) satisfies the inequality, while \(x = 3\), from \((2, 4)\), does not. Testing further with \(x = 5\) for \([4, \infty)\) confirms that it also satisfies the inequality. This systematic method helps in accurately determining the solution set for quadratic inequalities.