Problem 34 - College Algebra

Question

Graph the function. Observe the points of intersection and shade the \(x\) -axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation. $$ |x+3| \geq 5 $$

Step-by-Step Solution

Answer

The solution in interval notation is \((-\infty, -8] \cup [2, \infty)\).

Step 1: Understanding Absolute Value Inequality

We start with the inequality \(|x+3| \geq 5\). The absolute value inequality \(|a| \geq b\) implies two possible situations: \(a \geq b\) or \(a \leq -b\). Here, we set \(x + 3 \geq 5\) or \(x + 3 \leq -5\).

Step 2: Solve Each Inequality Separately

Let's solve \(x + 3 \geq 5\) first:Subtract 3 from both sides: \(x \geq 2\).Now, let's solve \(x + 3 \leq -5\):Subtract 3 from both sides: \(x \leq -8\).

Step 3: Determine the Solution Set

The solutions from Step 2 give us two separate intervals. For \(x \geq 2\), the interval is \([2, \infty)\). For \(x \leq -8\), the interval is \((-\infty, -8]\). Thus, the solution set for \(|x+3| \geq 5\) is the union of these intervals: \((-\infty, -8] \cup [2, \infty)\).

Step 4: Graph the Solution Set

On the number line, shade the interval \((-\infty, -8]\) to the left of -8 and \([2, \infty)\) to the right of 2. Use a closed dot at -8 and an open dot at 2, and shade toward negative infinity from -8 and toward positive infinity from 2. This shading represents the values of \(x\) that satisfy the original inequality.

Key Concepts

Interval Notation

Interval notation is a mathematical shorthand used to describe the set of solutions for an inequality. It allows for a concise representation of the parts of the number line that satisfy the conditions given. In the case of our exercise, we are dealing with two separate intervals as solutions:

• The first interval, \([2, \infty)\), includes all numbers greater than or equal to 2, and continues infinitely towards larger numbers.
• The second interval, \(( - \infty, -8] \), encompasses all numbers less than or equal to -8, extending infinitely in the negative direction.

The entire solution to our inequality \(|x+3| \geq 5\) is expressed as the union \(( - \infty, -8] \cup [2, \infty)\). The union symbol \(\cup\) indicates that any number in either interval satisfies the inequality. Using interval notation, we combine the segments of the solutions into a compact form, making it easier to interpret and visualize the range of possible numbers that meet the criteria of the inequality.

Graphing Inequalities

Graphing inequalities involves plotting the solutions to an inequality on a number line. This visual representation helps illustrate which parts of the number line include numbers that satisfy the inequality. In our situation, we have found two intervals resulting from dividing and solving the absolute value inequality \(|x+3| \geq 5\).To graph:

• Start with a number line and identify key points: -8 and 2.
• Above -8, place a closed dot to signify that -8 is part of the solution \(( - \infty, -8] \).
• Shade the line to the left of -8, extending towards negative infinity, to show all lesser numbers are included.
• At 2, place an open dot, indicating that 2 itself is included in the interval \([2, \infty)\).
• Shade the line towards positive infinity from 2, marking all greater numbers as part of the solution set.

This graph visually demonstrates the two regions where the inequality is true. It provides an immediate understanding of which values of \(x\) make the expression \(|x+3| \geq 5\) valid.

Absolute Value Functions

Absolute value functions represent the distance of a number from zero on the number line, regardless of direction. This means that both positive and negative numbers are treated as their positive counterparts. For example, \(|-7|\) and \(|7|\) both equal 7 because both are seven units away from zero.In equation form, the absolute value function \(|x+3| \geq 5\) can lead to two scenarios:

• The expression inside the absolute value, \(x + 3\), could be 5 or greater, giving rise to the inequality \(x + 3 \geq 5\).
• Alternatively, \(x + 3\) might be less than or equal to -5, forming the inequality \(x + 3 \leq -5\).

This dual nature stems from how absolute values can produce two possible conditions. Absolute value inequalities, like \(|x+3| \geq 5\), effectively break down into two separate, yet related inequalities. Solving these gives us the required intervals that form the solution set, helping us understand the possible values of \(x\) that meet the condition. This segmentation is crucial as it transforms a single complex inequality into manageable pieces.