Problem 66 - Elementary and Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. \(|x+5| \geq 7\)

Step-by-Step Solution

Answer

The solution is \((- fty, -12] \cup [2, fty)\).

Step 1: Understand the Inequality

In this exercise, we need to solve the inequality \(|x+5| \geq 7\). The absolute value inequality \(|A| \geq B\) means that either \(A \geq B\) or \(A \leq -B\). We will use this understanding to solve the given inequality.

Step 2: Break Absolute Value Inequality

To solve \(|x+5| \geq 7\), we break it into two separate cases: 1. \(x+5 \geq 7\)2. \(x+5 \leq -7\). Each case will be solved separately.

Step 3: Solve the First Inequality

For the inequality \(x+5 \geq 7\), subtract 5 from both sides to isolate \(x\):\[x \geq 2\]

Step 4: Solve the Second Inequality

For the inequality \(x+5 \leq -7\), subtract 5 from both sides to isolate \(x\):\[x \leq -12\]

Step 5: Combine Solutions

The original inequality \(|x+5| \geq 7\) is satisfied when either case is true. Therefore, the solution is:\(x \geq 2\) or \(x \leq -12\).In interval notation, this is written as:\((-fty, -12] \cup [2, fty)\).

Step 6: Graph the Solution Set

The graph of the solution set shows two intervals: for \(x \leq -12\), shade all values to the left of \(-12\), including \(-12\) itself (use a filled circle at \(-12\)), and for \(x \geq 2\), shade all values to the right of \(2\), including \(2\) itself (use a filled circle at \(2\)).

Key Concepts

Absolute Value

The concept of absolute value refers to the distance of a number from zero on a number line, regardless of direction. This means the absolute value always produces a non-negative number. For instance, the absolute value of both \(3\) and \(-3\) is \(3\). To understand absolute value inequalities such as \(|x+5| \geq 7\), it is important to consider the two possible scenarios: either the expression inside the absolute value is greater than or equal to 7 (\(x+5 \geq 7\)) or less than or equal to -7 (\(x+5 \leq -7\)). These two cases help break down the problem into manageable parts.
By setting up separate inequalities, we can solve each one independently. This approach reveals all possible solutions to the inequality and underscores the logic behind absolute value expressions in equations and inequalities.

Interval Notation

Interval notation is a way to describe a set of numbers along the number line. It clarifies which numbers are included in or excluded from an interval. Intervals are expressed using brackets and parentheses:

• A square bracket \([\text{ or } ]\) indicates inclusion of an endpoint.
• A round parenthesis \(\text{( or )}\) indicates exclusion of an endpoint.

When solving inequalities like \(|x+5| \geq 7\), the solution is described using interval notation. After isolating \(x\) in both scenarios and finding \(x \geq 2\) or \(x \leq -12\), we express this with intervals:

• \((-\infty, -12]\) for values satisfying \(x \leq -12\)
• \([2, \infty)\) for values satisfying \(x \geq 2\)

The union symbol (\(\cup\)) connects these intervals, representing the complete solution: \((-\infty, -12] \cup [2, \infty)\). This tells us the solution includes all numbers less than or equal to -12 or greater than or equal to 2.

Graphing Inequalities

Graphing inequalities on a number line is a visual way to represent the solutions. When working with inequalities such as \(x \geq 2\) or \(x \leq -12\), it helps to use a number line to clearly show the range of possible solutions.
Here's how you can graph these solutions:

• Draw a number line and mark the numbers \(-12\) and \(2\).
• Use a filled circle at \(-12\) to signify that this endpoint is included.
• Shade all values to the left of \(-12\) to indicate \(x \leq -12\).
• Similarly, place a filled circle at \(2\) and shade to the right of \(2\) to represent \(x \geq 2\).

This graphical representation clarifies the inequality solution by showing which portions of the number line satisfy the inequality conditions. It's a helpful tool, especially when dealing with more complex inequality systems, as it allows a quick check of solutions and their respective ranges.