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Chapter 2: Problem 12

For Problems \(1-16\), solve each equation. $$ |3 x-1|-1=9 $$

Short Answer

Expert verified
The solutions are \(x = \frac{11}{3}\) and \(x = -3\).

Step by step solution

01

Isolate the Absolute Value Expression

First, add 1 to both sides of the equation to isolate the absolute value expression: \(|3x - 1| = 10\.\)
02

Consider the Definition of Absolute Value

Recall that the absolute value equation \(|A| = B\) can be split into two separate equations: \(A = B\) and \(A = -B\). This yields: \(3x - 1 = 10\) and \(3x - 1 = -10\.\)
03

Solve the First Equation

Solve \(3x - 1 = 10\): Add 1 to both sides to get \(3x = 11\). Then, divide both sides by 3: \(x = \frac{11}{3}\.\)
04

Solve the Second Equation

Solve \(3x - 1 = -10\): Add 1 to both sides to get \(3x = -9\). Then, divide both sides by 3: \(x = -3\.\)
05

Write the Solution Set

The solutions to the equation are \(x = \frac{11}{3}\) and \(x = -3\). Therefore, the solution set is \(\left\{-3, \frac{11}{3}\right\}\.\)

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Key Concepts

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

Solving Equations
Solving equations is a fundamental skill in algebra. It involves finding the value of a variable that makes an equation true. In the context of absolute value equations, you're looking to balance both sides of the equation considering both the positive and negative scenarios.
This is because absolute values measure distance from zero and can be satisfied by both positive and negative numbers. To solve an absolute value equation like \(|3x - 1| - 1 = 9\), the first step is to isolate the absolute value expression. In our example, you would add 1 to both sides, resulting in \(|3x - 1| = 10\).
This simplification helps set the stage for deeper analysis.
Intermediate Algebra
Intermediate algebra is where students tackle more complex equations, including absolute value equations. Understanding the properties of absolute values is essential here, as these concepts help solve equations like \(|A| = B\).The main property to remember is:
  • If \(|A| = B\), then \(A = B\) and \(A = -B\).
In practice, this means that for each absolute value equation, there are two possible equations. For example, with \(|3x - 1| = 10\), you need to check both \(3x - 1 = 10\) and \(3x - 1 = -10\).
Solving these separately gives you the complete set of potential solutions.
Solution Sets
Once you've solved the individual equations derived from the absolute value, you'll have a solution set. This set contains all possible values of \(x\) that satisfy the original equation.From our earlier example steps:
  • Solving \(3x - 1 = 10\) gives \(x = \frac{11}{3}\).
  • Solving \(3x - 1 = -10\) results in \(x = -3\).
Thus, the solution set for the equation \(|3x - 1| - 1 = 9\) is \(\{-3, \frac{11}{3}\}\).
Solution sets are crucial as they summarize the possible solutions that make the original equation true.

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Most popular questions from this chapter

For Problems \(31-54\), solve each inequality. $$ |2 x-1| \leq 9 $$

For Problems \(17-30\), solve each inequality and graph the solution. $$ |x-2| \geq 1 $$

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -4<3-x<4 $$

Solve each inequality. $$ |1-2 x|<2 $$

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -6<4 x-5<6 $$
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