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Chapter 2: Problem 87

Write each as an equivalent absolute value equation. $$ x=6 \text { or } x=-6 $$

Short Answer

Expert verified
The equivalent absolute value equation is \(|x| = 6\).

Step by step solution

01

Understanding Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. Given expressions like \(x = 6\) or \(x = -6\), which are both 6 units away from zero, they can be represented using an absolute value.
02

Encoding as an Absolute Value Equation

To express \(x = 6\) or \(x = -6\) as an absolute value equation, note that both conditions imply the magnitude of \(x\) is 6, making the equation \(|x| = 6\).
03

Verifying the Equation

Check if \(|x| = 6\) aligns with the original equations. When \(|x| = 6\), \(x\) can be 6 or -6, thus covering both possibilities from the original \(x = 6\) and \(x = -6\).

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Key Concepts

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

Distance on the Number Line
Imagine a number line stretching indefinitely in both directions, with zero right in the middle. Each number on this line has a specific position relative to zero. This position is not only about which number it is, or whether it's positive or negative, but also about how far it is from zero. When we talk about distance on the number line, we're referring to this measure of how far a number is from zero, regardless of direction.

For example:
  • The number 6 is 6 units away from zero.
  • The number -6 is also 6 units away from zero.
Both are equidistant from zero, even though they lie on opposite sides. That's the beauty of absolute value 鈥� it treats both directions as equal because distance itself is always positive.
Magnitude of a Number
When we refer to the magnitude of a number, we're essentially talking about its size in terms of how far it is from zero on the number line. Sometimes, the term 'magnitude' is used interchangeably with 'absolute value.'

This can be represented by the symbol \(|x|\), where \(x\) is any number. Essentially, \(|x|\) captures the idea that we care only about the magnitude, not the direction.

For instance, \(|6|\) equals 6, and \(|-6|\) also equals 6. This is because magnitude considers only the distance from zero:
  • A positive number's magnitude is the number itself.
  • A negative number's magnitude is the number without the negative sign.
Understanding this concept is key in forming absolute value equations like \(|x| = 6\), which show us that \(x\) can be either 6 or -6 while maintaining the same magnitude.
Equivalent Equations
Equivalent equations are two or more equations that have the same solution set. They describe the same relationship between variables, even if they appear different upfront.

In our example, the original equations \(x = 6\) or \(x = -6\) describe the points on the number line that are 6 units away from zero. Both these conditions are captured by the absolute value equation \(|x| = 6\).

To illustrate:
  • The equation \(x = 6\) tells us that \(x\) is exactly 6 units to the right of zero.
  • The equation \(x = -6\) tells us that \(x\) is 6 units to the left of zero.
  • However, the equation \(|x| = 6\) encompasses both these possibilities in one single statement.
This unification of conditions into one equation is a core feature of equivalent equations, making them powerful in simplifying and solving mathematical problems.

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Most popular questions from this chapter

Solar system distances are so great that units other than miles or kilometers are often used. For example, the astronomical unit \((A U)\) is the average distance between Earth and the sun, or \(92,900,000\) miles. Use this information to convert each planet's distance in miles from the sun to astronomical units. Round to three decimal places. The planet Mercury's AU from the sun has been completed for you. (Source: National Space Science Data Center). $$ \begin{array}{|l|c|c|} \hline \text { Planet } & {\text { Miles from the Sun }} & {\text { AU from the Sun }} \\ \hline \text { Mercury } & {36 \text { million }} & {0.388} \\ \hline \end{array} $$ Venus \(\quad 67.2\) million

Solve each inequality. Graph the solution set and write it in interval notation. $$ |x-3|<8 $$

Write \(x>1\) or \(x<-1\) as an equivalent inequality containing an absolute value.

Solve each inequality. Graph the solution set and write it in interval notation. $$ \left|\frac{3}{5}+4 x\right|-6<-1 $$

Solve each inequality. Graph the solution set and write it in interval notation. $$ |5 x+2|<8 $$
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