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Chapter 9: Problem 26

Find the solution set for each equation. $$|3 x-2|+4=4$$

Short Answer

Expert verified
The solution set is {2/3}

Step by step solution

01

Clear terms outside the absolute value

Isolate the absolute value term on one side of the equation. To do this, subtract 4 from both sides of the equation: \[|3x - 2| + 4 - 4 = 4 - 4\] This simplifies to \[|3x - 2|= 0\]
02

Remove absolute value

The absolute value of a quantity is equal to zero if and only if the quantity itself is zero. Therefore, we can remove the absolute value and set the term inside equal to zero: \[3x - 2 = 0\] Then, solve the equation for x by adding 2 to both sides and dividing by 3: \[3x = 2\] so \[x = \frac{2}{3}\] The solution set therefore contains the single number \(\frac{2}{3}\) since an absolute value equal to zero should only have one solution.

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Key Concepts

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

Solution Set
When solving absolute value equations, we aim to find the set of values that satisfy the given equation. Such a set is known as the solution set. In algebra, the solution set is crucial because it tells us all possible answers to an equation.

For our exercise, |3x - 2| + 4 = 4, we start by isolating the absolute value expression |3x - 2| by subtracting 4 from both sides. This results in |3x - 2| = 0. Divide and conquer this into smaller parts:
  • Zero property: The equation implies that 3x - 2 is equal to zero.
  • Solve 3x - 2 = 0, leading to x = \( \frac{2}{3} \).
  • Result: The solution set is \( \{ \frac{2}{3} \} \).
The solution set of an absolute value equation can represent all possible x-values that make the equation true.
Algebra Problems
Algebra problems often involve manipulating equations to find unknown values. These problems require a strategic approach by using different algebraic principles.

In order to solve an absolute value equation like |3x - 2| + 4 = 4, one must follow several key steps:
  • Isolate: Separate the absolute value expression from other terms, simplifying the equation.
  • Evaluate: Understand that if the absolute value is zero, the term inside must also be zero.
  • Solve: Use algebraic operations to solve the resulting equation from Step 2.
Apply these strategies to not just understand individual problems but also develop a comprehensive skillset for tackling various types of equations in algebra.
Introductory Algebra
Introductory algebra serves as the foundation for understanding more complex mathematical concepts. It's where students learn to deal with expressions, equations, and basic operations.

Starting with simple linear equations and absolute value equations, students begin to grapple with more complex ideas:
  • Variables: Learners get comfortable with symbols like x which represent unknown numbers.
  • Operations: They execute basic operations such as addition, subtraction, multiplication, and division.
  • Expressions: Understanding expressions and combining like terms to simplify them is critical.
Through exercises like finding solution sets of absolute value equations, students build fundamental skills. Mastering these basics is essential, as they prepare students for more elaborate topics in mathematics.

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