Problem 68 - 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. $$|x|>5$$

Step-by-Step Solution

Answer

The solution for \(|x|>5\) is \((- \infty, -5) \cup (5, \infty)\)

Step 1: Solve the inequality without the absolute value symbol

Rewriting \(|x|>5\) without absolute value bars results in two inequalities: \(x>-5\) and \(x<5\).

Step 2: Find the solution set

Based on these inequalities, the solution set is all values of x that are either less than -5 or greater than 5.

Step 3: Graph the solution set on a number line

On a number line with marked points -5, and 5, shade everything to the left of -5, and everything to the right of 5.

Step 4: Express the solution set in interval notation

The interval notation for the solution is \((- \infty, -5) \cup (5, \infty)\)

Key Concepts

Interval Notation

Interval notation is a sleek way of expressing a range of values. It's particularly useful in inequalities, where you deal with collections of numbers rather than specific ones.
In this context, an interval represents all the numbers in between known as endpoints. These can be expressed as 'open' using parentheses

• Open intervals, like \((a, b)\), do not include the endpoints themselves, indicating values greater than \(a\) and less than \(b\).
• In contrast, closed intervals use brackets \[ [a, b] \] to include the endpoints.

For this particular inequality \(|x|>5\), we need to use open intervals, since \(|x|>5\) covers values less than -5 or more than 5, resulting in two separate intervals.Therefore, we express it as \((-\infty, -5) \cup (5, \infty)\), where \(\cup\) signifies the union of values from both intervals. It's a union of segments on either end of the number line, capturing all possible solutions.

Solution Set

The solution set is the collection of all possible solutions for the given inequality. When dealing with absolute value inequalities, it's common to break down the problem into simpler parts.
For \(|x|>5\):
The absolute value inequality translates into two situations.

• First inequality: \(x < -5\), representing values strictly less than -5.
• Second inequality: \(x > 5\), representing values strictly greater than 5.

Consequently, the solution set encompasses all numbers that satisfy either of these conditions. In our case, it's continuous in two segments, so we express it with set notation and interval notation.
In this problem, it means any possible number you pick, should either be less than -5 or greater than 5, where no points actually touch -5 or 5. This encompasses numbers from negative infinity up to -5 and from 5 onward to positive infinity.

Number Line

A number line is a visual tool that facilitates finding and illustrating the solution set derived from inequalities.
Imagine a straight line with numbers marked at regular intervals.

To graph the solution set of the inequality \(|x|>5\), plot:

• Open circles at the points -5 and 5, indicating these values themselves are not included in the solution set.
• Shade the entire region to the left of -5, which shows all numbers less than -5.
• Similarly, shade the region extending to the right of 5, which captures all numbers greater than 5.

This shaded visual representation reflects the solution intervals \((-\infty, -5)\) and \(5, \infty)\), helping you visually verify the plausible solutions where the inequality holds true.
Number lines are incredibly helpful in supporting your understanding of the range and scope of the solutions.