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Chapter 9: Problem 32

Solve each absolute value equation. \(|y|=-9\)

Short Answer

Expert verified
No solutions because the absolute value cannot be negative.

Step by step solution

01

Understanding Absolute Value

The absolute value of a number represents its distance from zero on the number line, and is always non-negative. Thus, \(|y|\) can never be negative, because absolute value signifies distance which can't be negative.
02

Analyze the Given Equation

The given equation is \(|y| = -9\). This requires that the absolute value of \(y\) equals \(-9\). However, since the number -9 is negative, no value of \(y\) satisfies the equation.

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Key Concepts

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

Absolute Value Properties
Absolute value might sound like a complex term, but it's quite simple! It tells us how far away a number is from zero on the number line. Imagine yourself picking numbers and measuring the distance to zero.

The catch is: absolute value is always non-negative. Distances don't have negative values. For example, the absolute value of 5 is written as \(|5| = 5\), and for -5, it's \(|-5| = 5\).

  • Absolute value 'flattens out' any negative signs. Think of it as always giving a positive version of the number's distance.
  • If you see an equation like \(|x| = a\), you need to remember this positive-only rule. It means x could travel \(a\) steps away from zero.
  • If \(a\) is negative, like in \(|x| = -9\), there鈥檚 no solution! Why? Negative distances don鈥檛 exist in our realm!
Solving Equations
Now you may wonder, how do we solve absolute value equations? There are some general approaches. Let's break it down:

  • If you have an equation \(|x| = a\), first ensure \(a\) is not negative. If it is negative, there is no solution, because absolute values can't be negative.
  • For equations like \(|x| = a\), simply understand this equation can split into two separate equations: \(x = a\) and \(x = -a\). This is because distance from zero can be in either direction on the number line.
  • After splitting, solve each of these basic equations separately to find all possible values of \(x\) that work.

In our exercise example, \(|y| - 9\), there simply isn鈥檛 a value of \(y\) that satisfies the equation because -9 isn鈥檛 a valid absolute value. This helps us finalize our solution with confidence.
Non-negative Distance Concept
In mathematics, the concept of distance is one of the vital fundamentals. Absolute value serves as the key idea that defines distance from zero on a number line and emphasizes its non-negative nature.

  • Imagine you are on a straight road, trying to figure out how far you are from a designated spot called 'zero'. Whether you are to the left of zero or to the right, your distance remains the same.
  • This non-negative distance is exactly what absolute value represents. It assures that no matter how you travel, your distance to zero cannot drop below zero!
  • In solving equations, it's essential to keep in mind that you can't have a negative absolute value result, similar to how you can't have a negative measure of distance.

This idea not only simplifies the number line but also reinforces the fact that absolute value equations need carefully checking for logically possible solutions, such as ensuring the right side of the equation is non-negative.

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Most popular questions from this chapter

Evaluate each expression for the given replacement value. See Section 5.1. \(x^{3}\) if \(x\) is \(-5\)

Graph the solution of each system of linear inequalities. See Examples 6 through 8. $$ \left\\{\begin{array}{l} {2 x+3 y<-8} \\ {x \geq-4} \end{array}\right. $$

Solve each compound inequality. Write solutions in interval notation. See Examples I through 8 . $$ x<\frac{2}{3} \text { or } x>-\frac{1}{2} $$

Solve each compound inequality. Write solutions in interval notation. See Examples 2 and \(3 .\) $$ -7 x \leq-21 \text { and } x-20 \leq-15 $$

Solve each compound inequality. See Examples 4 and 5. $$ 5
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