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Chapter 3: Problem 34

Solve and write interval notation for the solution set. Then graph the solution set. $$|x| \leq 4.5$$

Short Answer

Expert verified
The solution set in interval notation is \([-4.5, 4.5]\).

Step by step solution

01

- Understand the Absolute Value Inequality

Recall that an absolute value inequality of the form \( |x| \leq a \) can be rewritten as a compound inequality \( -a \leq x \leq a \).
02

- Rewrite the Given Inequality

Using the rule from Step 1, rewrite the given inequality \( |x| \leq 4.5 \) as \( -4.5 \leq x \leq 4.5 \).
03

- Write the Solution Set in Interval Notation

The compound inequality \( -4.5 \leq x \leq 4.5 \) can be expressed in interval notation as \([-4.5, 4.5]\).
04

- Graph the Solution Set

To graph the solution set \([-4.5, 4.5]\), draw a number line, plot points at -4.5 and 4.5, and shade the region between them, including the endpoints.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

absolute value
The absolute value of a number is its distance from zero on the number line. This means the absolute value is always a non-negative number. For any real number x, the absolute value is denoted as \(|x|\), and it is defined as:

\[|x| = \begin{cases} x & \text{if } x \text{ is } \text{greater or equal to } 0\-x & \text{if } x \text{is less than } 0\text.\begin|cases\]

For example, \(|3| = 3\) and \(|-3| = 3\). This means |3| and |-3| are both 3 units away from 0. When dealing with absolute value inequalities, this concept helps rewrite the inequality into a more manageable form.
compound inequality
A compound inequality involves two separate inequalities joined by 'and' or 'or'. For the given problem, we deal with 'and'. When you see an absolute value inequality like \(|x| \ \textless \ a \), it can be rewritten as:

\[ -a \textless x \textless a \]

This form is a compound inequality meaning that x is between -a and a. The original problem \(|x| \leq 4.5\) translates to:
\[ -4.5 \leq x \leq 4.5 \]

This tells us that any value of x, which is between -4.5 and 4.5 (inclusive), satisfies the inequality.
interval notation
Interval notation is a concise way to describe a set of numbers along a number line. It uses parentheses \( \) and brackets \[ \] to indicate whether endpoints are included (closed) or excluded (open). For example:

  • \( (a, b) \) - All numbers between a and b, but not including a and b.
  • \( [a, b] \) - All numbers between a and b, including both a and b.

In the context of our problem, the solution to \( -4.5 \leq x \leq 4.5 \) includes the endpoints -4.5 and 4.5. So, we use brackets to represent this interval:

\( [-4.5, 4.5] \)
number line graphing
Graphing an inequality on a number line visualizes the solution set. Here's how to graph \([-4.5, 4.5]\):

  • Draw a horizontal line, and mark points -4.5 and 4.5 on it.
  • Shade the region between -4.5 and 4.5.
  • Place a closed circle (or solid dot) on -4.5 and 4.5 to indicate these points are included in the solution set.

This shaded region and closed circles effectively show all values of x that satisfy the inequality \(-4.5 \leq x \leq 4.5\).

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Most popular questions from this chapter

In business, profit is the difference between revenue and cost; that is, $$ \begin{array}{l} \text { Total profit }=\text { Total revenue }-\text { Total cost, } \\ P(x)=R(x)-C(x) \end{array} $$ where \(x\) is the number of units sold. Find the maximum profit and the number of units that must be sold in order to yield the maximum profit for each of the following. $$R(x)=5 x, C(x)=0.001 x^{2}+1.2 x+60$$

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