Problem 61 - Algebra 2

Question

What is the complete solution to the equation \(|8-4 x|=40 ?\) F. \(x=8 ; x=12\) G. \(x=8 ; x=-12\) H. \(x=-8 ; x=-12\) J. \(x=-8 ; x=12\)

Step-by-Step Solution

Answer

Correct answer is J: \(x = -8\) and \(x = 12\).

Step 1: Understand the Definition of Absolute Value

Absolute value represents the distance of a number from zero on the number line, disregarding the direction. Thus, the expression \(|8-4x|=40\) signifies that the absolute value of the expression \(8-4x\) is 40.

Step 2: Set Up Two Equations

To eliminate the absolute value, set up two separate equations: one for the positive case and the other for the negative case. This will give us the situations \(8 - 4x = 40\) and \(8 - 4x = -40\).

Step 3: Solve the First Equation

Solve \(8 - 4x = 40\) by first subtracting 8 from both sides to get \(-4x = 32\). Then divide both sides by -4 to get \(x = -8\).

Step 4: Solve the Second Equation

Solve \(8 - 4x = -40\) by first subtracting 8 from both sides to get \(-4x = -48\). Then divide both sides by -4 to find \(x = 12\).

Step 5: Verify the Solutions

Substitute \(x = -8\) and \(x = 12\) back into the original equation \(|8-4x|=40\) to verify both satisfy the equation. Confirming this will show us the correct solution.

Step 6: Confirm the Correct Answer Choice

Our solutions \(x = -8\) and \(x = 12\) match answer choice J, confirming that \(x = -8\) and \(x = 12\) are the roots of the equation.

Key Concepts

Equation Solving

Solving equations, especially those involving absolute values, requires a solid understanding of how to manipulate and rearrange mathematical expressions. In the equation

• \(|8-4x|=40\)

this method begins with the recognition that absolute values create two potential scenarios—or solutions—stemming from their definition. Each scenario captures a different aspect of the possible solutions. The steps involved in this process generally include setting up two equations:

• one for the positive scenario *(i.e. without the absolute value symbol)* and,
• another for the negative scenario.

This is due to the fact that absolute value measures distance without regard to direction.To solve

• *the equation* \(|8-4x|=40\),
• start by setting up
  • \(8 - 4x = 40\)
  • and \(8 - 4x = -40\)

This will then allow you to explore both possibilities of the value \(8 - 4x\) and find the correct solutions for \(x\).

Absolute Value Definition

Understanding absolute value is pivotal in solving equations like \(|8-4x|=40\). Absolute value specifically refers to the distance a number is from zero on a number line, always expressed as a non-negative. In mathematical terms, the notation \(|a|\) represents that no matter the sign of \(a\), its absolute value is positive or zero. For example,

• \(|5| = 5\)
• \(|-5| = 5\)

So when dealing with \(|8-4x|=40\), we interpret the absolute value as implying two potential equations:

• \(8 - 4x = 40\)
• and \(8 - 4x = -40\).

These essences of absolute value help us break down the equation into simpler components that are more straightforward to solve.

Verification of Solutions

Once solutions \(x = -8\) and \(x = 12\) have been found, verifying them is crucial to ensure these numbers correctly solve the original equation \(|8-4x|=40\). This verification step involves substituting each solution back into the original equation to see if the absolute value holds true. For:

• \(x = -8\): substitute into \(|8 - 4(-8)|\), which evaluates to \(|8 + 32| = |40|\).
• For \(x = 12\): substitute into \(|8 - 4(12)|\), which evaluates to \(|8 - 48| = |-40|\).

As the absolute values |40| and |-40| both result in 40, verifying each solution satisfies the equation. Thus, the solutions \(x = -8\) and \(x = 12\) are correct and valid, making them the final answer.