Problem 56 - Elementary and Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. \(|x|<9\)

Step-by-Step Solution

Answer

The solution is \(-9 < x < 9\) or in interval notation: (-9, 9).

Step 1: Understand the Inequality

The inequality \(|x| < 9\) states that the absolute value of \(x\) must be less than 9. This means that \(x\) can take any value between -9 and 9, excluding -9 and 9 themselves.

Step 2: Write the Inequality without Absolute Value

To solve \(|x| < 9\), we rewrite it as a compound inequality: -9 < x < 9. This tells us that \(x\) must lie between -9 and 9.

Step 3: Graph the Solution on a Number Line

To graph the solution, draw a number line and place open circles at -9 and 9 because these values are not included in the solution. Shade the region between -9 and 9, representing all values \(x\) can take.

Step 4: Write the Solution using Interval Notation

Using interval notation, the solution is written as (-9, 9). The parentheses indicate that -9 and 9 are not part of the solution set.

Key Concepts

Understanding Absolute Value Inequalities

Absolute values measure the distance a number is from zero on the number line, without considering direction. In the inequality \(|x|<9\), we want the distance between the number \(x\) and zero to be less than 9. This doesn't tell us the exact value of \(x\), but rather that \(x\) is somewhere between -9 and 9, not touching either point.

To remove the absolute value, you rewrite it as a double inequality: \(-9 < x < 9\). This shows that \(x\) includes every value between these two numbers, but not the endpoints.

Remember, when solving inequalities with absolute values:

• \(|x|
• \(|x|>a\) transforms into two separate inequalities: \(x>a\) or \(x<-a\).

Each scenario has a different method, but understanding the core idea is crucial.

Graphing Inequality Solutions

Graphing is a great visual tool for understanding the range of values that solve an inequality. For our inequality \(-9 < x < 9\), think about a number line.

First, mark -9 and 9 with open circles, because these numbers are not part of the solution set. Open circles indicate that the endpoints are excluded. Then, shade the line in between these two points. This shaded region represents all the possible values that \(x\) could be.

• Always use open circles for inequalities like \(<\) and \(>\).
• Use closed circles when inequalities are \(\leq\) or \(\geq\).

Graphing transforms abstract problem-solving into something more tangible and easier to understand.

Using Interval Notation

Interval notation is a shorthand way to express the set of solutions for an inequality. It uses brackets and parentheses to describe the start and end points of the solution set.

For \(-9 < x < 9\), written in interval notation, it becomes \((-9, 9)\). The parentheses signal that -9 and 9 are not included in the solution.

• Use parentheses \(()\) to exclude endpoints from the solution set.
• Use brackets \([]\) to include endpoints in the solution set.

For example, if the inequality allowed \(x\) to be -9 or 9, it would be written as \([-9, 9]\) in interval notation. Understanding interval notation is important for clearly presenting solutions in a concise form.