Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, solve the equation involving absolute value. $$ |4 x+1|-3=6 $$

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = -\frac{5}{2}\).

Step 1: Isolate the Absolute Value

To solve the equation \(|4x + 1| - 3 = 6\), start by isolating the absolute value. Add 3 to both sides of the equation to get:\[ |4x + 1| = 9 \]

Step 2: Set Up Two Separate Equations

The expression \(|4x + 1| = 9\) means that the quantity inside the absolute value can be either 9 or -9.Therefore, we set up two separate equations:1. \(4x + 1 = 9\)2. \(4x + 1 = -9\)

Step 3: Solve the First Equation

Solve the equation \(4x + 1 = 9\):1. Subtract 1 from both sides: \(4x = 8\).2. Divide by 4: \(x = 2\).So, one solution is \(x = 2\).

Step 4: Solve the Second Equation

Solve the equation \(4x + 1 = -9\):1. Subtract 1 from both sides: \(4x = -10\).2. Divide by 4: \(x = -\frac{10}{4} = -\frac{5}{2}\).So, the second solution is \(x = -\frac{5}{2}\).

Step 5: Solution Verification

Verify both solutions in the original equation:1. For \(x = 2\): - \(|4(2) + 1| - 3 = |9| - 3 = 6\), which is true.2. For \(x = -\frac{5}{2}\): - \(|4(-\frac{5}{2}) + 1| - 3 = |-9| - 3 = 6\), which is also true. Both solutions satisfy the original equation.

Key Concepts

Solving Equations

To solve an equation involving an absolute value, such as \(|4x+1|-3=6\), the first step is understanding what the absolute value represents. In a mathematical equation, the absolute value is about determining the magnitude of a number, regardless of its sign. This gives the direction of solving an absolute value equation. You must look at both the positive and negative scenarios of the expression within the absolute value.
It's crucial to start with the right mindset as this influences how you interpret the problem. Think of the equation in two parts: the actual numerical expression and the operation involving the absolute value. The ultimate goal is to isolate the term inside the absolute value, work through the scenarios, and figure out the two potential solutions.

Step by Step Solution

When you are faced with an equation like \(|4x+1|-3=6\), a step by step approach offers clarity and preciseness:

• **Step 1:** Isolate the absolute value. First, add 3 to both sides which gives you \(|4x+1|=9\).
• **Step 2:** Understand the meaning. With \(|4x+1|=9\), consider that the expression inside can equal either 9 or -9.
• **Step 3:** Set up separate equations for the two scenarios: \(4x+1=9\) and \(4x+1=-9\).
• **Step 4:** Solve these equations to determine the values of \(x\).

By breaking the problem down into manageable parts, each step builds upon the previous, maintaining logical flow and structure. Let's now delve into solving the individual equations.

Solution Verification

Once solutions are found, as in this problem where \(x=2\) and \(x=-\frac{5}{2}\), it's important to verify them. This verification ensures that the solutions truly satisfy the original equation:

• For \(x=2\): Substitute back into the equation, \(|4(2)+1|-3=|9|-3=6\). The right side equals the left, showing this is a correct solution.
• For \(x=-\frac{5}{2}\): Substitute similarly, \(|4(-\frac{5}{2})+1|-3=|-9|-3=6\), and the equation balances, confirming this solution as well.

Verification assures you haven't made mistakes along the way by thoroughly checking that both solutions fit into the equation's framework. It solidifies your problem-solving reliability.

Isolation of Variables

A key step in solving equations with absolute values is isolating the variable term. This is crucial because it simplifies the equation, allowing you to apply further mathematical operations.
For the given problem \(|4x+1|-3=6\), isolation begins by handling the absolute constant. By adding 3, you set up the equation as \(|4x+1|=9\). This removes any external complications and focuses solely on the absolute expression.

• **Step 1:** Recognize that separating the constant externally from the equation simplifies the analysis.
• **Step 2:** Address the internal expression next, breaking it further down through setting and solving two distinct equations.

This clarity in isolating variables doesn't just apply to absolute value equations but is a fundamental technique used across various types of mathematical problems.