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Chapter 8: Problem 60

Solve. $$ |y-2|=|2-y| $$

Short Answer

Expert verified
The equation is true for all values of y.

Step by step solution

01

Identify Absolute Value Properties

Understand that the absolute value function \(|x|\) measures the distance of a number from zero on the number line, so it always results in a non-negative value.
02

Simplify the Equation

Recognize that \(|y-2|\) and \(|2-y|\) are equivalent because they both measure the distance between y and 2. Therefore, the equation \(|y-2| = |2-y|\) is always true for any y.
03

Confirm the Relationship

Since \(|y-2|\) is always equal to \(|2-y|\), any value of y will satisfy the original equation.

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Key Concepts

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

absolute value properties
The absolute value of a number is its distance from zero on a number line. This means that absolute value is always non-negative. For any real number, the absolute value is represented as \(|x|\). For example, \(|-3| = 3\) and \(|3| = 3\).

There are some key properties of absolute value to keep in mind:
  • \(|a| \geq 0\) : The absolute value of any number is never negative.
  • \(|-a| = |a|\): The absolute value of a number is the same as the absolute value of its opposite.
  • \(|a \cdot b| = |a| \cdot |b|\) : The absolute value of a product is the product of the absolute values.
  • \(|a/b| = |a|/|b|\) : The absolute value of a quotient is the quotient of the absolute values, provided that \(|b| 鈮� 0)\).
equivalent expressions
Equivalent expressions are expressions that have the same value, even if they look different. In the context of absolute value, it's essential to know that \(|a - b|\) is the same as \(|b - a|\). This is a crucial property used in solving equations involving absolute values.

Let's break it down:
  • Both \(|a - b|\) and \(|b - a|\) measure the distance between the numbers \(a\) and \(b\).
  • Distance is always a positive measure regardless of the direction, hence \(a\) and \(b\) can be switched without changing the result.
Recognizing these equivalent expressions can simplify solving absolute value equations like \(|y-2| = |2-y|\) because you can quickly see that the equation holds true for all y.
solving equations
When solving absolute value equations, you often need to consider different cases that satisfy the equation. Let's focus on the example, \(|y-2| = |2-y|\).

Start by simplifying the equation:
  • Recognize that \(|y-2|\) measures the distance between \(y\) and \(2\).
  • Also, \(|2-y|\) measures the distance between \(2\) and \(y\).
  • Given the properties of absolute values, \(|y-2|\) is indeed equivalent to \(|2-y|\).

This leads to the conclusion that any value of \(y\) will keep the equation true. You don't need more calculations because understanding these properties makes it clear. This approach showcases how certain absolute value equations can be simplified significantly using their inherent properties.

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Most popular questions from this chapter

For \(F(x)=\frac{1}{x-3},\) find all \(x\) -values for which \(F(x) \leq 2\).

Solve. $$ \frac{1}{3} x^{3}-x+\frac{2}{3}>0 $$

A \(\$ 6.00\) toll is charged to cross the bridge from mainland Florida to Sanibel Island. A six-month reduced-fare pass, costing \(\$ 50.00\) reduces the toll to \(\$ 2.00 .\) A six-month unlimited-trip pass costs \(\$ 300\) and allows free crossings. How many crossings in six months does it take for the reducedfare pass to be the more economical choice?

For \(f(x)=14-x^{2},\) find all \(x\) -values for which \(f(x)>5\).

Solve by graphing. $$ x-1=-5 $$
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