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Chapter 9: Problem 13

Solve each absolute value equation. \(|2 x|=0\)

Short Answer

Expert verified
The solution is x = 0.

Step by step solution

01

Understand Absolute Value Property

The absolute value expression \(|2x|\) represents the distance of \(2x\) from 0 on a number line. The equation \(|2x| = 0\) tells us that the distance from zero is zero, meaning \(2x\) itself must be zero.
02

Set the Absolute Expression to Zero

Since \(|2x| = 0\), we conclude that \(2x = 0\). This is because only zero has an absolute value of zero.
03

Solve for x

To solve for \(x\), divide the equation \(2x = 0\) by 2. This simplifies to \(x = 0\). Thus, the solution to the equation is \(x = 0\).

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Key Concepts

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

Distance on a Number Line
When we talk about distance on a number line, we are referring to how far a number is from another number, often zero. In mathematical terms, this distance is represented by absolute values. For example, the absolute value of a number like 5, or written as \(|5|\), signifies that 5 is exactly 5 units away from 0.
  • Absolute values are always non-negative, as distance cannot be negative.
  • If an absolute value is zero, it indicates that the number is exactly at the origin, or zero itself.
In our exercise, we were given the equation \(|2x| = 0\). This means we need to find the point where the quantity \(2x\) has zero distance from 0. Therefore, \(2x = 0\) is itself zero.
Zero Property of Absolute Value
The zero property of absolute value is a fundamental concept that states: If the absolute value of a number is zero, then the number itself must be zero.
  • This is because distance can only be zero when the point or number is exactly at the reference point.
  • With absolute values, this reference point is zero. Hence, if \(|a| = 0\), it implies \(a = 0\).
For instance, in the equation \(|2x| = 0\), we apply the zero property to determine that \(2x\) must equal zero. That's because no other number, apart from zero, can have a zero absolute value. This fundamental understanding sets the stage for solving the equation.
Solving Linear Equations
Solving linear equations often involves isolating the variable we want to find, typically represented by \(x\) in the equations. Let's break down the process using the step from our example where \(2x = 0\).
  • First, identify the term containing the variable, which is \(2x\).
  • Then, perform operations to remove any coefficients or terms connected to \(x\).
Here, the operation involves dividing both sides of the equation by 2 to solve for \(x\).
Thus, \(\frac{2x}{2} = \frac{0}{2}\), which simplifies to \(x = 0\). The solution is straightforward because the zero property made it easy to isolate \(x\), showcasing how absolute value equations can simplify into simple linear equations when the zero property is applied.

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Most popular questions from this chapter

A word processor charges 22 dollars per hour, \(x\), for typing a first draft, and 15 dollars per hour, \(y,\) for making changes and typing a second draft. If you need a document typed and have 100 dollars, the inequality $$ 22 x+15 y \leq 100 $$ represents your situation. Graph the inequality in the first quadrant only.

Solve each compound inequality. Write solutions in interval notation. See Examples 2 and \(3 .\) $$ -7 x \leq-21 \text { and } x-20 \leq-15 $$

Solve each compound inequality. Write solutions in interval notation. See Examples I through 8 . $$ -2 x<-6 \text { or } 1-x>-2 $$

Find by inspection all values for \(x\) that make equation true. See Sections 1.2 and 1.4. $$ |x|=0 $$

Solve each compound inequality for x. See the example below. To solve x-6<3 x<2 x+5, notice that this inequality contains a variable not only in the middle, but also on the left and the right. When this occurs, we solve by rewriting the inequality using the word and. CAN'T COPY THE GRAPH $$ 7 x-1 \leq 7+5 x \leq 3(1+2 x) $$
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