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Chapter 9: Problem 7

Find the solution set for each equation. $$\left|\frac{4 x-2}{3}\right|=2$$

Short Answer

Expert verified
The solution set for the given equation is \(x = 2\) and \(x = -1\).

Step by step solution

01

Create two separate equations

Since the absolute value equation can be both positive and negative, we need to isolate the absolute value on one side of the equation and then create two separate equations. The given equation is \( \left|\frac{4 x-2}{3}\right|=2 \). This can be transformed into \( \frac{4x - 2}{3} = 2 \) and \( \frac{4x - 2}{3} = -2 \).
02

Solve the first equation

Let's solve the first equation \( \frac{4x - 2}{3} = 2 \). To do this we isolate for x by first multiplying the entire equation by 3, resulting in \(4x - 2 = 6\). Then, we add 2 to both sides, yielding \(4x = 8\). Lastly, divide both sides by 4 which gives \(x = 2\).
03

Solve the second equation

Now let's solve the second equation \( \frac{4x - 2}{3} = -2 \). Isolate for x by multiplying the entire equation by 3, which gives \(4x - 2 = -6\). Add 2 to both sides to get \(4x = -4\). Lastly divide both sides by 4 to achieve \(x = -1\).

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Key Concepts

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

Equation Solving
When solving an equation that involves absolute values, it's essential to understand that the absolute value of a number is its distance from zero on a number line. This means it can be either positive or negative but always results in a non-negative value. In equations like \( \left| \frac{4x-2}{3} \right| = 2 \), we treat the absolute value expression as two potential scenarios.

These scenarios are:
  • Positive expression: \( \frac{4x-2}{3} = 2 \)
  • Negative expression: \( \frac{4x-2}{3} = -2 \)
This dual approach helps capture both possibilities of the absolute value.

The primary goal is to create equations that are easy to work with, thus breaking down the absolute value equation into two separate and simpler equations.
Algebraic Manipulation
Algebraic manipulation involves rearranging equations to solve for the variable in question. Begin by focusing on isolating the variable. This often entails performing arithmetic operations such as adding, subtracting, multiplying, or dividing.

For the equation \( \frac{4x - 2}{3} = 2 \), we multiply through by 3 to eliminate the fraction, yielding \( 4x - 2 = 6 \). By adding 2 to both sides, we move towards isolating \( x \):
  • Addition: \( 4x - 2 + 2 = 6 + 2 \) simplifies to \( 4x = 8 \)
  • Division: Dividing both sides by 4 results in \( x = 2 \)
Similarly, solving \( \frac{4x - 2}{3} = -2 \) involves the same clear steps. These operations ensure that \( x \) is isolated effectively, giving the clear answer.
Solution Sets
After performing equation solving and algebraic manipulation, it's time to form the solution set. This includes all possible solutions derived from the equations.

From our two separate equations, we found:
  • \( x = 2 \) from \( \frac{4x - 2}{3} = 2 \)
  • \( x = -1 \) from \( \frac{4x - 2}{3} = -2 \)
Thus, the solution set for the original absolute value equation \( \left| \frac{4x-2}{3} \right| = 2 \) is \( \{2, -1\} \).

Solution sets compile every possible solution of an equation, representing all values that satisfy the given equation. Understanding this helps in recognizing the complete range of valid answers for mathematical problems.

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Most popular questions from this chapter

Write the given sentences as a system of linear inequalities in no variables. Then graph the system. The sum of the \(x\) -variable and the \(y\) -variable is at most 4 The \(y\) -variable added to the product of 3 and the \(x\) -variable does not exceed 6

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I'm considering the compound inequality \(x<8\) or \(x \geq 8\) so the solution set is \((-\infty, \infty)\)

. Explain how to graph the solution set of a system of inequalities.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. It's not a good thing for a business if \(R(x)>C(x)\)

The ordered pair \((-2,40)\) satisfies the following system: $$ \left\\{\begin{array}{c} y \geq 9 x+11 \\ 13 x+y>14 \end{array}\right. $$
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