Problem 60 - 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 \((-\infty, -7] \cup [-3, \infty)\).

Step 1: Understanding the Absolute Value Inequality

We begin by interpreting the inequality \(|x+5| \geq 2\). This inequality means that the distance between \(x+5\) and 0 is at least 2. An absolute value inequality \(|a| \geq b\) translates to two simultaneous inequalities: \(a \geq b\) or \(a \leq -b\).

Step 2: Solving the Inequality \(x+5 \geq 2\)

First, we solve the inequality \(x+5 \geq 2\). To do this, subtract 5 from both sides: \[x + 5 - 5 \geq 2 - 5\] which simplifies to \(x \geq -3\).

Step 3: Solving the Inequality \(x+5 \leq -2\)

Next, we solve the inequality \(x+5 \leq -2\). Subtract 5 from both sides: \[x + 5 - 5 \leq -2 - 5\] which simplifies to \(x \leq -7\).

Step 4: Combining the Solutions

From the two inequalities, the solutions are \(x \geq -3\) and \(x \leq -7\). Combining these inequalities correctly, the set of solutions is \(x \leq -7\) or \(x \geq -3\). These are disjoint intervals: one is for \(x \geq -3\), and the other is for \(x \leq -7\).

Step 5: Expressing as Intervals

The solutions can be expressed as intervals: \(x \leq -7\) corresponds to \((-\infty, -7]\), and \(x \geq -3\) corresponds to \([-3, \infty)\). Therefore, the solution in interval notation is \((-\infty, -7] \cup [-3, \infty)\).

Step 6: Illustrating on the Number Line

To represent these intervals on a number line, use a filled circle at \(-7\) indicating inclusion of this endpoint, and a filled circle at \(-3\). Shade everything to the left of \(-7\) and everything to the right of \(-3\), indicating the regions where \(x\) satisfies the inequality.

Key Concepts

Real Numbers

Real numbers are basically all the numbers you encounter in everyday life. This includes whole numbers, fractions, decimals, and irrational numbers.

• Whole numbers: like 0, 1, 2, 3, ...
• Fractions: such as 1/2, -3/4
• Decimals: like 0.75 or -2.5
• Irrational numbers: such as the square root of 2 or Pi (π)

The concept of real numbers is essential when solving inequalities because they form the foundation of the number line. When you solve inequalities, you're usually finding which real numbers satisfy the condition of the inequality. In this case, the exercise looks into identifying the real numbers that fit the inequality \(|x+5| \geq 2\). These are particular values that lie within certain intervals on the number line.

Interval Notation

Interval notation is a method of writing subsets of the real number line. It's a concise way of describing a range of possible values that satisfy an inequality.In interval notation:

• A round bracket ( ) means the end point is not included.
• A square bracket [ ] means the endpoint is included.

For the inequality \(|x+5| \geq 2\), the solution \((x \leq -7) \cup (x \geq -3)\) is expressed in interval notation as \((-abla, -7] \cup [-3, abla)\), where:

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

This notation helps quickly identify the segments of the number line that contain the solutions.

Number Line Representation

The number line is a visual representation of real numbers in order. This visual aid is helpful when working with inequalities because it clearly shows which numbers satisfy the conditions.When you depict intervals like those from the inequality \(|x+5| \geq 2\), you would do the following on the number line:

• Place a filled circle at -7 and -3 to indicate these points are included in the solution.
• Shade the area to the left of -7, indicating all numbers less than or equal to -7 are solutions.
• Shade the area to the right of -3, showing that all numbers greater than or equal to -3 fulfill the inequality.

This number line representation makes complex sets of solutions easier to visualize.

Disjoint Intervals

Disjoint intervals are segments on the number line that do not overlap. They represent separate ranges of real numbers satisfying a condition.In our case, the inequality \(|x+5| \geq 2\) leads to solutions comprising disjoint intervals: \((-abla, -7]\) and \[[-3, abla)\]. These intervals:

• Do not intersect each other.
• Represent distinct parts of the real number line - one part with numbers smaller than or equal to -7, and another part with numbers equal to or greater than -3.

Understanding disjoint intervals is crucial because it shows how the solutions branch out in different sections of the number line, demonstrating the diverse set of solutions for inequalities involving absolute values.