Problem 64 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\((-\infty, -7) \cup (7, +\infty)\)

Step 1: Understand the Absolute Value Notation

The inequality \(|x| > 7\) means that the distance of \(x\) from 0 on a number line is greater than 7. This can lead us to two separate cases because absolute value considers both sides of the number line.

Step 2: Set Up Two Cases

Since \(|x| > 7\), this can be rewritten as two separate inequalities: \(x > 7\) and \(x < -7\). These correspond to the two sides of 0 on the number line where the absolute value of \(x\) would exceed 7.

Step 3: Solve Each Inequality

Solve both inequalities separately:1. For \(x > 7\), the solution is any \(x\) greater than 7.2. For \(x < -7\), the solution is any \(x\) less than -7.

Step 4: Graph the Solution Set

On a number line, represent the solutions: 1. Draw an open circle at 7 and shade all the numbers greater than 7 to the right. 2. Draw an open circle at -7 and shade all the numbers less than -7 to the left.

Step 5: Write the Solution Using Interval Notation

Combine the two solution sets in interval notation:Since \(x > 7\) corresponds to \((7, \, \infty)\) and \(x < -7\) corresponds to \((-\infty, \, -7)\), the combined solution is written as the union of the two intervals:\((-\infty, \, -7) \cup (7, \, \infty)\).

Key Concepts

Understanding Absolute Value

Absolute value is a concept in mathematics used to describe the distance a number is from zero on the number line, without considering its direction. It is always non-negative because distance cannot be negative. For any real number, the absolute value is represented as \(|x|\). For example, \(|3| = 3\) and \(|-3| = 3\), indicating both numbers are three units away from zero.
When dealing with inequalities involving absolute values, such as \(|x| > 7\), it means we are looking for the set of all numbers whose distance from zero is greater than 7. This results in two separate conditions: \((x > 7)\) or \((x < -7)\). These arise because both positive and negative numbers can be a certain distance from zero.

• If \(x\) is greater than 7, the numbers of interest are to the right of 7 on a number line.
• If \(x\) is less than -7, the numbers of interest are to the left of -7 on a number line.

Graphing Inequalities on a Number Line

Graphing inequalities helps us to visualize the set of solutions. When dealing with inequalities like \(|x| > 7\), you’re concerned with what happens on a number line. Here’s how you can graph these solutions:

• For \(x > 7\), draw an open circle around the number 7 to indicate that 7 itself is not included in the solution. Shade everything to the right towards infinity, which represents numbers greater than 7.
• For \(x < -7\), draw an open circle around -7. Then, shade to the left towards negative infinity to represent numbers less than -7.

This visual representation shows that the solution consists of two parts: \(x\) values greater than 7 and \(x\) values less than -7. Open circles indicate that the end numbers are not part of the solution.

Using Interval Notation for Inequalities

Interval notation is a mathematical shorthand used to express which parts of the number line are included in the solution set. For inequalities, it is a concise way to represent multiple ranges at once. With our example, \(|x| > 7\), we are interested in values less than -7 and greater than 7.

• The interval \((7, \, \infty)\) describes all numbers greater than 7. The parenthesis indicates that the endpoint 7 is not included, and the symbol \(\infty\) means it extends indefinitely to the right.
• Similarly, the interval \((-\infty, \, -7)\) describes all numbers less than -7, extending indefinitely to the left.

When combining these intervals using a union due to the \(|x| > 7\) condition, it becomes \((-\infty, \, -7) \cup (7, \, \infty)\). The union symbol \((\cup)\) indicates that solutions can belong to either range. This unified interval reflects that the solutions exclude the range between -7 and 7. Interval notation simplifies writing and understanding the solution set as it effectively communicates which numbers satisfy the inequality.