Problem 31 - College Algebra

Question

For the following exercises, solve the equation involving absolute value. $$ |1-4 x|-1=5 $$

Step-by-Step Solution

Answer

Solutions are \(x = -\frac{5}{4}\) and \(x = \frac{7}{4}\).

Step 1: Isolate the Absolute Value

To solve the equation \(|1-4x|-1=5\), first isolate the absolute value. Add 1 to both sides:\[|1-4x| = 6\]

Step 2: Set Up Two Equations

The equation \(|1-4x| = 6\) implies two possible scenarios: 1. The expression inside the absolute value is positive: \(1-4x = 6\).2. The expression inside the absolute value is negative: \(1-4x = -6\).

Step 3: Solve for x (Positive Scenario)

Solve the equation where the expression is considered positive:\[1 - 4x = 6\]Subtract 1 from both sides to get:\[-4x = 5\]Then, divide both sides by -4:\[x = -\frac{5}{4}\]

Step 4: Solve for x (Negative Scenario)

Solve the equation where the expression is considered negative:\[1 - 4x = -6\]Subtract 1 from both sides:\[-4x = -7\]Then, divide both sides by -4:\[x = \frac{7}{4}\]

Step 5: Verify the Solutions

Check both solutions in the original equation \(|1-4x|-1=5\):- For \(x = -\frac{5}{4}\): \[|1-4(-\frac{5}{4})| = |1+5| = |6| - 1 = 5\] which is correct.- For \(x = \frac{7}{4}\): \[|1-4(\frac{7}{4})| = |1-7| = |-6| - 1 = 5\] which is correct.

Key Concepts

Algebraic Solutions

Algebraic solutions for equations involve finding the exact values of variables that satisfy the equation. In the case of absolute value equations, this means identifying the values that satisfy both the main equation and any derived equations when considering the nature of absolute values. Absolute value functions return non-negative numbers, so they often result in two potential equations to solve.

Let's consider an absolute value equation such as \( |1-4x|-1=5 \). By isolating the absolute value, you simplify the equation to \( |1-4x| = 6 \). At this point, you set up two separate equations to solve for the unknown variable:

• The positive scenario: \( 1-4x = 6 \)
• The negative scenario: \( 1-4x = -6 \)

Both scenarios must be addressed to find potential solutions as either could be true because absolute value represents distance from zero, which can be either positive or negative in origin.

Verification of Solutions

Verification of solutions is a crucial step in any equation-solving process. It ensures that the solutions satisfy the original equation. For absolute value equations, this step involves substituting the potential solutions back into the original equation to check their validity.

For the solutions \( x = -\frac{5}{4} \) and \( x = \frac{7}{4} \), derived from the absolute value equation \( |1-4x|-1=5 \), verification means checking if these values result in the original equation holding true.

• For \( x = -\frac{5}{4} \): Substitute in the original equation to yield \[ |1 - 4(-\frac{5}{4})| - 1 = 5 \]
  This simplifies to \[ |6| - 1 = 5 \] which holds true.
• For \( x = \frac{7}{4} \): Substitute in the original equation to yield \[ |1 - 4(\frac{7}{4})| - 1 = 5 \]
  This simplifies to \[ | -6 | - 1 = 5 \] which also holds true.

Verification not only confirms correctness but also boosts confidence in the solution process undertaken.

Step by Step Math Solutions

The beauty of step by step math solutions lies in breaking down complex problems into manageable parts. This method is very effective for solving absolute value equations. By approaching the problem one step at a time, students can see each part of the equation being dealt with systematically.

For the equation \( |1-4x|-1=5 \), the step-by-step approach involves:

• Isolating the absolute value: Add 1 to both sides to get \( |1-4x| = 6 \).
• Setting up two equations: Handle both the positive \( 1-4x = 6 \) and the negative \( 1-4x = -6 \) scenarios.
• Solving for x: Solve each equation to find possible solutions: \( x = -\frac{5}{4} \) and \( x = \frac{7}{4} \).
• Verification: Substitute solutions back into the original equation to check their validity.

This methodical breakdown not only makes problem-solving easier, it also builds strong foundations, improving the students' problem-solving skills for future challenges.

Equation Solving Methods

Equation solving methods vary depending on the type of equation. Absolute value equations leverage the property that the absolute value of a number is always non-negative, which often leads to two separate cases to consider when solving.

With \( |1-4x| = 6 \), the initial task is to isolate the absolute value expression. Once achieved, you then consider the two cases derived from the absolute value:

• Scenario 1: The expression inside equals the given value (positive scenario).
• Scenario 2: The expression inside equals the negative of the given value (negative scenario).

By considering these two scenarios, you account for all possible values that make the absolute value equal to the target number. These methods build on the fundamental idea that solving equations is about finding all possible solutions that satisfy the given conditions.