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Chapter 3: Problem 18

Answer the given questions. Where are all points such that \(|x|=2 ?\)

Short Answer

Expert verified
The points are at x = 2 and x = -2.

Step by step solution

01

Understand the Absolute Value

The expression |x| = 2 implies the distance of  x from 0 is 2 on the number line. Absolute value measures the magnitude regardless of direction.
02

Solve the Absolute Value Equation

Since absolute value represents distance,  |x| = 2 states that  x can either be 2 or -2. This means  x = 2 or  x = -2.

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Key Concepts

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

Solving Absolute Value Equations
Solving absolute value equations seems tricky at first, but it's all about understanding the nature of absolute value. The absolute value of a number is its distance from zero, regardless of its direction on the number line. This means that
  • both positive and negative numbers have the same absolute value if they are the same distance from zero.
  • for example, the absolute value of 2 is 2, and the absolute value of -2 is also 2.
When solving equations like \(|x| = 2\), we want to find all the numbers that are exactly two units away from zero. Since distance on the number line can be measured in both directions, the solutions for \(|x| = 2\) are both \(x = 2\) and \(x = -2\). Hence, in solving absolute value equations, we should look for both the positive and the negative values that can satisfy the equation.
Number Line
A number line is a visual representation that helps in understanding numbers and their absolute values. It is a horizontal line where numbers are placed at intervals. Zero is typically in the center, with positive numbers to the right and negative numbers to the left. This makes it easy to see the distance from zero:
  • Numbers located at an equal distance from zero but in opposite directions have the same absolute value.
  • For \(|x| = 2\), both 2 and -2 are on the number line at two units away from zero.
You can use a number line to visually verify solutions to absolute value equations by checking the distances from zero to ensure they match the required distance given in the equation. The number line thus serves as an excellent tool for visualizing and solving absolute value problems.
Distance from Zero
When we talk about absolute value, we are essentially discussing the concept of distance from zero. It measures how far a number is from zero, ignoring any direction. Distance is always a non-negative value, as it cannot be less than zero. Let's break it down:
  • A positive number and its negative counterpart both have the same distance from zero, hence the same absolute value.
  • For example, \(|2| = 2\) and \(|-2| = 2\), both numbers are 2 units away from zero.
  • This makes sense when we think of a real-world distance, which is always positive, like how many steps you take.
So, in equations involving absolute value, you can think of it as finding which numbers have a specific distance from zero, as determined by the equation. This understanding simplifies analyzing and solving such equations greatly.

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Most popular questions from this chapter

Solve the given problems. A balloon is being blown up at a constant rate. (a) Sketch a reasonable graph of the radius of the balloon as a function of time. (b) Compare to a typical situation that can be described by \(r=\sqrt[3]{3 t},\) where \(r\) is the radius (in \(\mathrm{cm}\) ) and \(t\) is the time (in \(\mathrm{s}\) ).

Graph the indicated functions. On a taxable income of \(x\) dollars, a certain city's income \(\operatorname{tax} T\) is defined as \(T=0.02 x\) if \(020,000 .\) Graph \(T=f(x)\) for \(0 \leq x<100,000\).

Graph the indicated functions. A copper electrode with a mass of \(25.0 \mathrm{g}\) is placed in a solution of copper sulfate. An electric current is passed through the solution, and \(1.6 \mathrm{g}\) of copper is deposited on the electrode each hour. Express the total mass \(m\) on the electrode as a function of the time \(t\) and plot the graph.

Graph the indicated functions. The voltage \(V\) across a capacitor in a certain electric circuit for a \(2-s\) interval is \(V=2 t\) during the first second and \(V=4-2 t\) during the second second. Here, \(t\) is the time (in s). Plot \(V\) as a function of \(t\).

A function and how it is to be shifted is given. Find the shifted function and then display the given function and the shifted function on the same screen of a graphing calculator. \(y=\sqrt{x},\) right 3
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