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Chapter 1: Problem 21

Solve the equations. \(2=8+|11 y+4|\)

Short Answer

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Step by step solution

01

Isolate the Absolute Value Expression

Start by isolating the absolute value expression on one side of the equation. Begin by subtracting 8 from both sides of the equation: \[ 2 - 8 = |11y + 4| \] Simplify the left side: \[ -6 = |11y + 4| \]
02

Consider the Absolute Value Definition

Recall that the absolute value of any expression is always non-negative. Since \(-6\) is negative, there is no solution to the equation because an absolute value can never equal a negative number.

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Key Concepts

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

Isolate the Absolute Value
In solving absolute value equations, the first step is to isolate the absolute value expression. This means you need to get the absolute value term by itself on one side of the equation. This is crucial because it sets the stage for applying absolute value properties and solving the equation. In the example given, we start with the equation:
2 = 8 + |11y + 4|.
Our goal is to isolate |11y + 4|. To do this, begin by subtracting 8 from both sides:
2 - 8 = |11y + 4|.
Simplify the left side:
-6 = |11y + 4|.
Now, the absolute value term |11y + 4| is isolated, making it easier to evaluate and solve.
Absolute Value Properties
Absolute value properties are fundamental to understanding absolute value equations. Recall that the absolute value of a number is its distance from 0 on a number line, regardless of direction. This distance is always non-negative. For any expression inside the absolute value, this principle holds true:
|A| 鈮� 0, where A is any expression.
In our example, after isolating |11y + 4|, we get:
-6 = |11y + 4|.
Notice that -6 is a negative number. According to the absolute value property, an absolute value expression cannot equal a negative number because its result is always non-negative. This property instantly tells us something important about the solution.
No Solution Condition
There are certain conditions under which an absolute value equation has no solution. Specifically, if you isolate the absolute value expression and find that it equals a negative number, then the equation has no solution. In our scenario, we ended up with:
-6 = |11y + 4|.
Since the absolute value |11y + 4| is always non-negative, it cannot be equal to -6. Therefore, we conclude that there is no solution to the equation. This is known as the 'no solution condition'.
The steps to determine this are:
  • Isolate the absolute value term.
  • Check if the isolated absolute value term equals a negative number.
  • If it does, then there's no solution to the equation.
Understanding this condition is crucial for solving absolute value equations effectively.

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