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Chapter 1: Problem 111

Solve each equation or inequality. $$ |5 x-2|=0 $$

Short Answer

Expert verified
x = \(\frac{2}{5}\)

Step by step solution

01

- Understand the Absolute Value

The absolute value equation \(|5x - 2| = 0\) requires an understanding that the absolute value of a number is always non-negative and equals the number itself when it is zero.
02

- Set the Expression inside the Absolute Value to Zero

For the absolute value to be zero, the expression inside must be zero. Thus, set the expression inside the absolute value equal to zero: \[ 5x - 2 = 0 \]
03

- Solve for x

Solve the equation \(5x - 2 = 0\) to find the value of \(x\). Add 2 to both sides to isolate the term with \(x\): \[ 5x = 2 \] Next, divide both sides by 5: \[ x = \frac{2}{5} \]

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Key Concepts

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

Solving Equations
When solving equations, the main goal is to find the value of the variable that makes the equation true. This involves several steps: understanding the equation, manipulating it using mathematical operations, and isolating the variable.

In the given exercise, we have the equation \(|5x - 2| = 0\). To solve this, we need to consider the specific properties of absolute values and proceed accordingly. Remember, solving an equation step by step helps to ensure every part of the process is clear and accurate.
Absolute Value Properties
The absolute value of a number represents its distance from zero on the number line. It is always a non-negative number. Here are some key points about absolute values:
  • The absolute value of zero is zero: \(|0| = 0\).
  • The absolute value of a positive number is the number itself: \(|a| = a\) if \(a \geq 0\).
  • The absolute value of a negative number is the positive version of the number: \(|a| = -a\) if \(a < 0\).


In the problem \(|5x - 2| = 0\), we are using the fact that an absolute value equals zero only when the expression inside the absolute value is zero. This leads us to the next step of setting \(5x - 2 = 0\).
Isolation of Variable
The isolation of the variable is a crucial step in solving equations. The goal is to get the variable by itself on one side of the equation. Here鈥檚 how we do it for the equation \(5x - 2 = 0\):
  • Add or subtract terms: First, we add 2 to both sides to get: \(5x - 2 + 2 = 0 + 2\), resulting in \(5x = 2\).
  • Divide or multiply terms: Next, divide both sides by 5: \(x = \frac{2}{5}\).


Now, \(x\) is isolated and we have the solution \(x = \frac{2}{5}\). By isolating the variable, we determine the exact value of \(x\) that satisfies the original equation. This step-by-step approach ensures that we maintain the equation's balance and accurately solve for the variable.

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Most popular questions from this chapter

The 10 tallest buildings in Houston, Texas, are listed along with their heights. $$ \begin{array}{|l|c|} \hline \quad {\text { Building }} & \text { Height (in feet) } \\ \hline \text { JPMorgan Chase Tower } & 1002 \\ \text { Wells Fargo Plaza } & 992 \\ \text { Williams Tower } & 901 \\ \text { Bank of America Center } & 780 \\ \text { Texaco Heritage Plaza } & 762 \\ \text { 609 Main at Texas } & 757 \\ \text { Enterprise Plaza } & 756 \\ \text { Centerpoint Energy Plaza } & 741 \\ \text { 1600 Smith St. } & 732 \\ \text { Fulbright Tower } & 725 \\ \hline \end{array} $$ Use this information To find the average of a group of numbers, we add the numbers and then divide by the number of numbers. Use a calculator to find the average of the heights.

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