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Chapter 9: Problem 8

Find the solution set for each equation. $$\left|\frac{3 x-1}{5}\right|=1$$

Short Answer

Expert verified
The solution set for the absolute value equation \(\left|\frac{3 x-1}{5}\right|=1\) is \(\{2,-\frac{4}{3}\}\).

Step by step solution

01

Setting Up the Two Equations

Because the number inside the absolute value (which is referred to as \(x\)) can be either negative or positive while keeping the same absolute value, it's necessary to set up two separate equations without absolute value. In this case those equations are: \[\frac{3x-1}{5} = 1\] and \[\frac{3x-1}{5} = -1\]
02

Solving the First Equation

Solving the first equation \(\frac{3x-1}{5} = 1\), first multiply both sides of the equation by 5 for removing the fraction to get: \(3x - 1 = 5\). Adding 1 to both sides gives: \(3x = 6\). Then divide by 3 to solve for \(x\), this gives: \(x = 2\).
03

Solving the Second Equation

Solving the second equation \(\frac{3x-1}{5} = -1\), first apply the same steps as in the first equation to get rid of the fraction: \(3x - 1 = -5\). When 1 is added to both sides: \(3x = -4\). And then when both sides are divided by 3, the final result for \(x\) in this case is: \(x = -\frac{4}{3}\).

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Key Concepts

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

Solving Equations
When solving absolute value equations, we are trying to find the values of the variable that make the equation true. Absolute values measure a number's distance from zero, ignoring any negative sign. For example, the absolute value of both 3 and -3 is 3.

This means that if you have an equation like \( \left|\frac{3x-1}{5}\right| = 1 \), you need to consider two scenarios:
  • The expression inside the absolute value equals 1: \( \frac{3x-1}{5} = 1 \).
  • The expression inside the absolute value equals -1: \( \frac{3x-1}{5} = -1 \).
By solving these two different equations, you can find every possible solution for the original absolute value equation.
Fraction Operations
Fraction operations might seem tricky at first, but they follow straightforward steps. In the context of solving equations, your goal is often to 'clear' the fraction to make calculations easier.

Here's how it's done:
  • See the equation \( \frac{3x-1}{5} = 1 \) as an example.
  • The fraction \( \frac{3x-1}{5} \) can be eliminated by multiplying every term of the equation by 5.
  • This leaves you with a simpler equation: \( 3x - 1 = 5 \).
  • Repeat the same process for equations involving a negative number.
What's crucial is maintaining balance by performing the same operation on both sides of the equation. This keeps the equality valid, leading you smoothly to solve for \( x \).
Solution Set
The solution set of an equation is simply the set of all values that satisfy the equation. For absolute value equations, such as \( \left|\frac{3x-1}{5}\right|=1 \), you might find more than one solution, as seen in this exercise.

From solving the two standard equations:
  • First, for \( 3x - 1 = 5 \), we found \( x = 2 \).
  • Second, for \( 3x - 1 = -5 \), we found \( x = -\frac{4}{3} \).
Thus, the solution set consists of \( x = 2 \) and \( x = -\frac{4}{3} \). This set tells us all the values for \( x \) that make the original equation true. By checking each solution individually in the context of the problem, you can verify they indeed satisfy the absolute value condition.

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