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Chapter 1: Problem 54

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x-4|<4$$

Short Answer

Expert verified
The solution is the interval \((0, 8)\).

Step by step solution

01

Understand the Inequality

The inequality \(|x-4| < 4\) implies that the absolute value of \(x - 4\) is less than 4, meaning the distance between \(x\) and 4 on the number line is less than 4.
02

Remove Absolute Value

The expression \(|x-4| < 4\) can be rewritten as two inequalities without the absolute value: \(x - 4 < 4\) and \(x - 4 > -4\). This gives us two separate conditions to solve.
03

Solve Each Inequality

- For \(x - 4 < 4\), adding 4 to both sides gives us \(x < 8\).- For \(x - 4 > -4\), adding 4 to both sides gives us \(x > 0\).
04

Combine Both Inequalities

The solution set is the interval where both inequalities \(x > 0\) and \(x < 8\) are satisfied simultaneously. In interval notation, this is written as \((0, 8)\).
05

Plot on Number Line

On a number line, draw a line segment from 0 to 8, using open circles at both endpoints to signify that 0 and 8 are not included in the interval. This visually represents the interval \((0, 8)\).

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Key Concepts

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

Interval Notation
Interval notation is a way to express a range of values, typically real numbers, in a compact form. It is especially useful in representing solutions to inequalities. In interval notation, a pair of numbers is used to indicate a range of real numbers between them. Symbols like parentheses \(\) and brackets \[\] help define whether the endpoints are included in the interval.
- Parentheses \(a, b\) indicate that the endpoints \(a\) and \(b\) are not included.- Brackets \[a, b\] show that the endpoints are included.
For the inequality \(|x-4|<4\), the solution set in interval notation is \(0, 8\), meaning it includes all real numbers between 0 and 8, but not 0 or 8 themselves.
Number Line Representation
A number line is a simple and effective visual tool to represent intervals and inequalities. You draw a horizontal line, marking relevant points like 0 and 8, which serve as boundaries for the interval. In our case, the interval solution \(0, 8\) is represented on the number line as a segment between 0 and 8. Use open circles at these points to indicate they are not part of the interval.
This format helps provide a quick visual understanding of solutions and is especially useful when dealing with multiple inequalities or complex expressions.
Solving Inequalities
Solving inequalities involves finding all possible values of a variable that make the inequality true. With absolute value inequalities like \(|x-4|<4\), the goal is to isolate the variable to find its possible range. Here, removing the absolute value splits the inequality into two simpler forms:
  • \(x-4<4\)
  • \(x-4>-4\)
Solving these gives \(x<8\) and \(x>0\), respectively. The solution of the inequality is found by identifying the overlap or intersection of these individual solution sets. In interval notation, this intersection becomes \(0, 8\).
Real Numbers
Real numbers encompass a wide range of numerical values including all integers, fractions, and decimals. They are represented on the number line, which extends infinitely in both directions. Understanding the concept of real numbers and their placement on the number line is crucial for solving inequalities.
These numbers allow us to precisely define intervals and distances, as in the example \(|x-4|<4\), which indicates a set of real numbers where the distance from \(x\) to 4 is less than 4. Thus, comprehending real numbers enhances our ability to solve and interpret inequality solutions efficiently.

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Most popular questions from this chapter

Find an equation for the line having the given slope and passing through the given point. Write your answers in the form \(y=m x+b\). (a) \(m=22 ;\) through (0,0) (b) \(m=-222 ;\) through (0,0)

This exercise outlines a proof of the fact that two nonvertical lines with slopes \(m_{1}\) and \(m_{2}\) are perpendicular if and only if \(m_{1} m_{2}=-1 .\) In the following figure, we've assumed that our two nonvertical lines \(y=m_{1} x\) and \(y=m_{2} x\) intersect at the origin. [If they did not intersect there, we could just as well work with lines parallel to these that do intersect at \((0,0),\) recalling that parallel lines have the same slope.] The proof relies on the following geometric fact: \(\overline{O A} \perp \overline{O B} \quad\) if and only if \(\quad(O A)^{2}+(O B)^{2}=(A B)^{2}\). (a) Verify that the coordinates of \(A\) and \(B\) are \(A\left(1, m_{1}\right)\) and \(B\left(1, m_{2}\right)\) (b) Show that $$\begin{aligned}O A^{2} &=1+m_{1}^{2} \\\O B^{2} &=1+m_{2}^{2} \\\A B^{2} &=m_{1}^{2}-2 m_{1} m_{2}+m_{2}^{2} \end{aligned}$$ (c) Use part (b) to show that the equation $$O A^{2}+O B^{2}=A B^{2}$$ is equivalent to \(m_{1} m_{2}=-1\) (GRAPH CAN'T COPY)

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x-3| \leq 4$$

Rewrite each statement using absolute value notation, as in Example 5.The distance between \(x\) and 1 is \(1 / 2\).

Graph the equation after determining the \(x\) - and \(y\) -intercepts and whether the graph possesses any of the three types of symmetry described on page 58 $$y=-1 / x^{3}$$
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