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Chapter 0: Problem 100

Solve the absolute value equation. $$3-|y+9|=11-3|y+9|$$

Short Answer

Expert verified
The solutions are \(y = -5\) and \(y = -13\).

Step by step solution

01

Isolate the Absolute Value

Start by bringing all terms involving \(|y+9|\) to one side of the equation.\[ 3 - |y+9| = 11 - 3|y+9| \]Move the terms:\[ -|y+9| + 3|y+9| = 11 - 3 \]This simplifies to:\[ 2|y+9| = 8 \]
02

Simplify the Equation

Divide each side by 2 to solve for the absolute value:\[ |y+9| = \frac{8}{2} \]Which simplifies to:\[ |y+9| = 4 \].This tells us that the expression inside the absolute value can be either 4 or -4.
03

Solve for Two Cases

Now, solve for \(y+9 = 4\) and \(y+9 = -4\).**Case 1:**\[ y+9 = 4 \]Subtract 9 from both sides:\[ y = 4 - 9 \rightleftharpoons y = -5 \]**Case 2:**\[ y+9 = -4 \]Subtract 9 from both sides:\[ y = -4 - 9 \rightleftharpoons y = -13 \]

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Key Concepts

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

Solving Equations
When it comes to solving equations, it is all about finding the value of the variable that makes the equation true. Here, you deal with an equation involving absolute values. This can seem tricky at first, but breaking it down into steps makes it more manageable. - Start by observing the structure of your equation and identify any absolute value expressions. - Next, aim to isolate these, so you can work with simpler expressions. The ultimate goal is to find all possible values for the variable that satisfy the equation. Remember, each step is a move closer to finding these solutions. Solving an equation is like solving a puzzle where each move is important and must follow logically from the previous one.
Isolating Variables
Isolation of variables is a fundamental step in solving equations, particularly when absolute value expressions are involved.- Begin by moving all terms containing the variable to one side of the equation. - For absolute value equations, you'll often need to perform operations to combine like terms. For example, in the equation you started with, terms involving \( |y+9| \) were brought to one side to simplify the process. Through isolation, you aim to have an expression like \( |variable| = constant \) which can then be split into more manageable cases. Remember, the idea is to make the equation as simple as possible so you can thoroughly analyze it.
Absolute Value Properties
Understanding absolute value properties is crucial for tackling equations that include them. Absolute values indicate a number's distance from zero on the number line, regardless of direction.- This means \( |x| = a \) can translate to two different potential equations: \( x = a \) or \( x = -a \). - When handling an absolute value equation, you need to consider both positive and negative results for the expression inside the absolute value.For example, when \( |y+9| = 4 \), it means that \( y+9 \) can either be \( 4 \) or \( -4 \). Recognizing this property is what allows you to solve for the variable by setting up and solving two distinct cases. Each case will lead to a potential solution for the problem.

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Most popular questions from this chapter

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