Problem 67 - College Algebra

Question

Solve each inequality in Exercises 57-84 by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$|x|>3$$

Step-by-Step Solution

Answer

The solution is \(x > 3\) or \(x < -3\). The interval notation for the solution set is \((-∞,-3) ∪ (3,∞)\).

Step 1: Rewrite the inequality without absolute value bars

Remember that for any number 'a', |x| > a means x > a or x < -a. Our inequality is |x| > 3, so it can be equivalently written as x > 3 or x < -3.

Step 2: Graph the solutions on a number line

On the number line, all numbers greater than 3 or less than -3 will be in the solution set. This look like two rays, starting at 3 and -3 and extending to the right and left respectively, without including the endpoints.

Step 3: Express the solution set in interval notation

Interval notation for x > 3 is (3,∞). For x < -3 it is (-∞,-3). Because both conditions are valid for the solution set, both intervals should be included and separated by a union. So the final solution set in interval notation is (-∞,-3) ∪ (3,∞).

Key Concepts

Interval Notation

Interval notation is a way to describe a set of numbers, or a "solution set." It helps us easily express the range of values that solve an inequality. When you see an interval written with parentheses, like \(3, \infty\), it means all numbers greater than 3 but not including 3 itself. This is often represented using parentheses, which denote that the endpoint is not part of the solution set, called "open intervals."

Here's a quick reference for interval notation:

• Use parentheses \( ( \, ) \) for numbers not included
• Use brackets \[ [ \, ] \] for numbers included, which we call "closed intervals"
• The symbol \( \infty \) or \( -\infty \) represents infinity, indicating the set goes on forever in that direction
• A union symbol \( \cup \) combines multiple intervals into one solution set

In the given inequality \(|x| > 3\), the solution in interval notation is \((-\infty, -3) \cup (3, \infty)\), indicating values less than -3 or greater than 3.

Number Line

A number line is a visual representation of numbers placed in order along a straight line. It's a helpful tool to graph inequalities and visualize solution sets. When dealing with an inequality, the number line shows which numbers satisfy the inequality.

In the example of \(|x| > 3\), the number line solution involves plotting two "rays":

• One ray starts just beyond -3 and extends leftward to represent \(x < -3\)
• The other starts just beyond 3, extending rightward to represent \(x > 3\)
• Empty circles are placed at -3 and 3 to show these points are not included (open intervals)

This illustration helps you see that any number on these rays will satisfy the inequality, giving a clear picture of the solution.

Inequalities Without Absolute Value

When solving absolute value inequalities, it's crucial to first rewrite them without the absolute value bars. This converts the absolute inequality into a set of two simple inequalities. Understanding this step is key to solving such problems.

For \(|x| > 3\), the absolute value statement implies two scenarios:

• \(x > 3\) - any number greater than 3 satisfies this part
• \(x < -3\) - any number less than -3 satisfies this part

The key takeaway is that absolute value inequalities often translate into a dual condition. This means that the solution should consider both parts independently and then be combined to find the overall solution set.

Solution Set in Algebra

The solution set in algebra refers to all the possible values of the variable that satisfy an inequality or equation. In inequalities, this is usually a range of numbers rather than a single number.

For the inequality \(|x| > 3\), the solution set combines values from two different ranges:

• Values that are greater than 3, expressed as \(3, \infty\)
• Values that are less than -3, expressed as \(-\infty, -3\)

Both these intervals together form the complete solution set, written as \((-\infty, -3) \cup (3, \infty)\).

This combined set of values meets the inequality conditions, covering all numbers satisfying the given algebraic inequality. It shows how algebraic notation helps us understand which numbers work to solve the equation or inequality accurately.