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Chapter 9: Problem 2

Solve each absolute value equation. \(|y|=15\)

Short Answer

Expert verified
The solutions are \(y = 15\) and \(y = -15\).

Step by step solution

01

Understanding Absolute Value

The absolute value of a number represents its distance from 0 on the number line, without considering direction. Therefore, if \(|y| = 15\), the solutions are the numbers whose distance from 0 is 15.
02

Setting Up the Equation

To solve \(|y| = 15\), recognize that \(y\) can be equal to 15 or -15, because both numbers satisfy the distance condition. Thus, the possible solutions are: \(y = 15\) and \(y = -15\).
03

Solution Verification

Check each value to verify they satisfy the original equation. For \(y = 15\), we have \(|15| = 15\), which is true. For \(y = -15\), \(|-15| = 15\), which is also true.

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Key Concepts

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

Distance from Zero
The absolute value of a number is a measure of its distance from zero on a number line. This concept is crucial because it ignores the direction, which means whether the number is positive or negative, the distance is always positive. For example:
  • The absolute value of 5 is 5, as it is five units away from zero.
  • The absolute value of -5 is also 5, again because it is five units away from zero.
Absolute value is denoted by vertical bars, such as \(|x|\), with the understanding being that the distance from zero never changes based on the sign of the number. This perspective is foundational when working with absolute value equations as it guides us to consider both positive and negative solutions.
Number Line
A number line is incredibly useful when understanding absolute value. It's a visual tool that allows us to see numbers laid out in order, both positive and negative, with zero in the center. When working with absolute value problems, you think of numbers in terms of how far they are from zero.
  • When you look at a number line, a number to the right of zero is positive.
  • A number to the left of zero is negative.
  • The distance between any number and zero is represented by its absolute value.
For instance, when solving \(|y|=15\), you'll locate zero on a number line, count 15 units to the right for the positive solution (15), and 15 units to the left for the negative solution (-15). This visualization reinforces the concept that distance from zero is what matters.
Solving Equations
Solving absolute value equations is about finding numbers that satisfy the given distance from zero. The equation \(|y|=15\) is a classic example. Here's how it's done:
  • First, identify the absolute value equation: \(|y|=15\).
  • Recognize that this means y could be either 15 or -15, because both numbers are 15 units from zero.
  • Set up the two possible equations: \y = 15\ and \y = -15\.
  • Solve each equation separately to find y. Both solutions are valid since they satisfy the absolute value condition.
Finally, verify each solution by plugging back into the original equation to check correctness. For both 15 and -15, the verification shows that their absolute value equals 15. This confirms our understanding that absolute value equations rely on both positive and negative roots.

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Most popular questions from this chapter

Graph the solution of each system of linear inequalities. See Examples 6 through 8. $$ \left\\{\begin{array}{l} {2 x+5 y \leq-10} \\ {y \geq 1} \end{array}\right. $$

Determine whether \((1,1)\) is included in each graph. See the Concept Check in this section. $$ y>5 x $$

Solve each compound inequality. Write solutions in interval notation. See Examples I through 8 . $$ 0<\frac{5-2 x}{3}<5 $$

Find by inspection all values for \(x\) that make equation true. See Sections 1.2 and 1.4. $$ |x|=7 $$

Graph the solution of each system of linear inequalities. See Examples 6 through 8. $$ \left\\{\begin{array}{l} {x \geq-3} \\ {y \geq-2} \end{array}\right. $$
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