Problem 59 - Precalculus

Question

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x-5| \geq 2$$

Step-by-Step Solution

Answer

The solution is the intervals \((-\infty, 3] \cup [7, \infty)\) on the number line.

Step 1: Understand the Absolute Value Inequality

The given inequality is \(|x-5| \geq 2\). This implies that the distance between \(x\) and 5 on the number line is at least 2. In other words, \(x\) can be either 2 or more units to the right of 5, or 2 or more units to the left of 5.

Step 2: Break Down the Inequality

Rewrite the inequality \(|x-5| \geq 2\) as two separate inequalities.\[x - 5 \geq 2 \quad \text{or} \quad x - 5 \leq -2\]These represent the two cases where the condition holds.

Step 3: Solve the First Inequality

Solve the inequality \(x - 5 \geq 2\). Add 5 to both sides to find:\[x \geq 7\]This means one portion of the solution set is all numbers \(x\) greater than or equal to 7.

Step 4: Solve the Second Inequality

Solve the inequality \(x - 5 \leq -2\). Add 5 to both sides to find:\[x \leq 3\]This indicates the other portion of the solution set is all numbers \(x\) less than or equal to 3.

Step 5: Combine Solutions into Intervals

The two solution sets can be combined into two intervals:\(x \leq 3\) is the interval \((-\infty, 3]\).\(x \geq 7\) is the interval \([7, \infty)\).The complete solution is the union of these two intervals: \((-\infty, 3] \cup [7, \infty)\).

Key Concepts

Interval Notation

Interval notation is a method used to describe a set of numbers along the number line. It captures the beginning and end of intervals where the inequality holds true. For inequalities, where expressions are either greater than, less than, or between values, interval notation provides a concise way to represent these conditions.

For instance, in the original exercise, we determined that the solution to \(|x-5| \geq 2\) resulted in two separate intervals: one where \((x \leq 3)\) and another where \((x \geq 7)\). In interval notation, these are written as:

• \((\-\infty, 3]\) — all numbers less than or equal to 3.
• \([7, \infty)\) — all numbers greater than or equal to 7.

The solution set of the inequality is presented as the union of these intervals, which is \((\-\infty, 3] \cup [7, \infty)\). The union symbol \((\cup)\) signifies all the numbers that belong to either of these intervals. Using interval notation helps organize and simplify expressing solution sets in mathematics.

Solving Inequalities

To solve absolute value inequalities, you first need to understand what the absolute value represents. The absolute value \(|x - a|\) indicates the distance between the number \(x\) and another point \(a\) on the number line. Solving \(|x - 5| \geq 2\) means finding all values of \(x\) where the distance from 5 is at least 2.

Breaking down the inequality involves setting up two conditions because the absolute value function is always non-negative.

• For \(x - 5 \geq 2\), solve for \(x\) to get \(x \geq 7\). This reveals numbers that are 2 units or more to the right of 5.
• For \(x - 5 \leq -2\), solve for \(x\) to get \(x \leq 3\). This provides numbers that are 2 units or more to the left of 5.

The solutions to these inequalities give us the intervals that form the complete solution set. By combining them, you obtain the final solution in interval notation. The two resulting inequalities help ensure that \(x\) satisfies the original condition of being outside the range between 3 and 7.

Number Line Representation

A number line is a visual tool used to represent numbers and their relationships. When solving inequalities, drawing a number line can help you see where the solution sets lie and how they relate to each other. For absolute value inequalities like \(|x-5| \geq 2\), the number line becomes an especially valuable resource.

Start by plotting critical numbers from the solved inequalities; in this case, 3 and 7. These numbers indicate where the solution set begins or ends. Then shade the regions that satisfy each inequality:

• Shade all numbers less than or equal to 3, moving leftwards, indicating the interval \((\-\infty, 3]\).
• Shade all numbers greater than or equal to 7, moving rightwards, showing the interval \([7, \infty)\).

The number line thus provides a clear representation of how the solution comprises two separate regions, helping you understand the solution's layout intuitively. This visualization assists in verifying the intervals noted down and ensures that you have captured the essence of the inequality appropriately.