Problem 7 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

x = 5 or x = -5

Step 1 - Understanding the Absolute Value

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

Step 2 - Split the Absolute Value Equation

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

Step 3 - 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: \[ x = \frac{-20}{4} = -5 \].

Key Concepts

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’s 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.