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Chapter 1: Problem 7

Solve each equation. $$ |4 x|=20 $$

Short Answer

Expert verified
x = 5 or x = -5

Step by step solution

01

- Understanding the Absolute Value

The absolute value expression \( |4x| \) means that the result of \( 4x \) can be either positive or negative.
02

- Split the Absolute Value Equation

Since \( |4x| = 20 \), this means \( 4x = 20 \) or \( 4x = -20 \). Write these two separate equations: \( 4x = 20 \) and \( 4x = -20 \).
03

- Solve Each Equation

Solve \( 4x = 20 \) by dividing both sides by 4: \[ x = \frac{20}{4} = 5 \]. Solve \( 4x = -20 \) by dividing both sides by 4: \[ x = \frac{-20}{4} = -5 \].

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Key Concepts

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

Absolute Value
Absolute value represents the distance a number is from zero on a number line, regardless of direction. In the context of an equation like \(|4x| = 20\), the absolute value expression \(|4x|\) shows that we consider both positive and negative scenarios of \(|4x|\) being equal to 20. This means \(|4x| = 20\) translates into two possible equations: \(4x = 20\) and \(4x = -20\). By splitting the absolute value equation into these two scenarios, we cover all potential solutions.
Linear Equations
Linear equations involve expressions that set a straight-line relationship between variables. For the absolute value equation \(|4x| = 20\), splitting it into \(4x = 20\) and \(4x = -20\) results in two simple linear equations. We solve these kinds of equations by isolating the variable. Let's see how:
  • For \(4x = 20\): Divide both sides by 4 to get \(x = 5\).
  • For \(4x = -20\): Divide both sides by 4 to get \(x = -5\).
Both equations are straightforward to solve, thanks to their linear nature.
Problem-Solving
Problem-solving is key when dealing with mathematical equations like these. Here鈥檚 a step-by-step strategy to help:
  • Identify the type of equation: Recognize if it's an absolute value, linear, quadratic, etc.
  • Understand the properties: Know what absolute value means and how it affects the equation.
  • Break it down: For absolute value equations, always split it into two separate linear equations.
  • Solve individually: Tackle each linear equation separately by isolating the variable.
  • Verify solutions: Make sure both solutions satisfy the original equation.
By following these steps, solving complex problems becomes manageable and logical.

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Most popular questions from this chapter

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