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Chapter 9: Problem 3

Solve each absolute value equation. \(|3 x|=12.6\)

Short Answer

Expert verified
The solutions are \(x = 4.2\) and \(x = -4.2\).

Step by step solution

01

Understand Absolute Value

The absolute value of a number is the distance of the number from zero, regardless of direction. Thus, for the equation \(|3x| = 12.6\), we need to solve for both positive and negative cases of the expression inside the absolute value.
02

Set Up Two Equations

The equation \(|3x| = 12.6\) can result from two scenarios: either \(3x = 12.6\) or \(3x = -12.6\). Therefore, we can write two separate equations: 1. \(3x = 12.6\) 2. \(3x = -12.6\)
03

Solve the First Equation

For the equation \(3x = 12.6\), divide both sides by 3 to isolate \(x\). This gives:\[x = \frac{12.6}{3} = 4.2\]
04

Solve the Second Equation

For the equation \(3x = -12.6\), divide both sides by 3 to isolate \(x\). This gives:\[x = \frac{-12.6}{3} = -4.2\]

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Key Concepts

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

Solving Equations
Solving absolute value equations involves finding all possible values that satisfy the equation. The core idea is to consider both positive and negative scenarios, since absolute value measures distance from zero and not direction. When tasked with an equation like \(|3x| = 12.6\), it translates to finding values of \(x\) that make \(3x = 12.6\) or \(3x = -12.6\).
  • The key step is setting up two separate equations, removing the absolute value operator.
  • This results in two straightforward linear equations to solve.
After setting up, solving the equations involves basic algebraic manipulations, typically division in this case, to isolate \(x\) and find both potential solutions.
Algebraic Solutions
Algebraic solutions are methods used to solve equations by manipulating the terms to isolate the variable of interest. When dealing with an equation like \(3x = 12.6\), a crucial operation is division. By dividing both sides by 3, we simplify the expression to solve for \(x\).
  • This step-by-step process involves applying the same operation to both sides of the equation to preserve equality.
  • Perform these operations carefully to avoid errors.
This precise method leads us to find one solution as \(x = 4.2\). For the alternate case of \(3x = -12.6\), performing the same operation gets \(x = -4.2\). Always double check calculations to ensure accuracy.
Absolute Value Properties
Understanding the properties of absolute values is key to solving these equations. The absolute value of a number refers to its distance from zero on the number line. This means that both positive and negative values have the same absolute value. For example, both 12.6 and -12.6 will have an absolute value of 12.6.
  • This property allows us to transform an equation like \(|3x| = 12.6\) into two separate equations without the absolute value.
  • It's essential to consider both positive and negative solutions in these scenarios.
Such understanding ensures that no potential solutions are overlooked, allowing you to correctly resolve any absolute value equation.

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Most popular questions from this chapter

Graph the solution of each system of linear inequalities. See Examples 6 through 8. $$ \left\\{\begin{array}{l} {2 x-5 y \leq 9} \\ {y \leq-3} \end{array}\right. $$

Graph the solution of each system of linear inequalities. See Examples 6 through 8. $$ \left\\{\begin{array}{l} {2 x+5 y \leq-10} \\ {y \geq 1} \end{array}\right. $$

Match each inequality with its graph. a. \(x>2\) b. \(y<2\) c. \(y<2 x\) d. \(y \leq-3 x\) e. \(2 x+3 y<6\) f. \(3 x+2 y>6\) (GRAPH CANNOT COPY)

Graph the solution of each system of linear inequalities. See Examples 6 through 8. $$ \left\\{\begin{array}{l} {x \leq 2} \\ {y \geq-3} \end{array}\right. $$

Find by inspection all values for \(x\) that make equation true. See Sections 1.2 and 1.4. $$ |x|=-2 $$
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