Problem 44 - Intermediate Algebra

Question

Solve each equation. See Example 3. $$ -10|16 x+4|-3=-3 $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{1}{4}\).

Step 1: Isolate the Absolute Value

Before we can work with the absolute value itself, we need to isolate it on one side of the equation. The original equation is \[-10|16x+4|-3=-3.\] First, add 3 to both sides to get: \[-10|16x+4| = 0.\] Now, divide by -10 to isolate the absolute value: \[|16x+4| = 0.\]

Step 2: Solve the Absolute Value Equation

With the absolute value isolated, the equation to solve is \[|16x+4| = 0.\] Since the absolute value of a number is zero only when that number itself is zero, set \[16x + 4 = 0.\] Solve for \(x\) by subtracting 4 from both sides: \[16x = -4.\] Then divide by 16: \[x = -\frac{4}{16} = -\frac{1}{4}.\]

Key Concepts

Isolation of Absolute Value

In solving equations involving absolute values, the first crucial step is to isolate the absolute value expression. Think of it like peeling an onion - we must remove the outer layers to reach the center.

Let's break down the process using the given equation:

• Start with the equation \[-10|16x+4|-3=-3.\]
• To isolate the absolute value \(|16x+4|\), first eliminate the additional terms around it. Add 3 to both sides, which simplifies the equation to \[-10|16x+4| = 0.\]
• Now, divide both sides by -10 to get rid of the coefficient attached to the absolute value. This results in \(|16x+4| = 0.\)

By isolating the absolute value, we focus directly on the core of the equation. This method simplifies our work and ensures accuracy.

Equation Solving Methods

After isolating the absolute value, the next step involves solving the equation that emerges. Understanding how absolute values function is key.

The absolute value equation \(|16x+4| = 0\) implies that the expression inside the absolute value is zero:

• Set the equation \(16x+4 = 0\) since absolute value zeroes out only with a zero inside.
• To solve for \(x\), subtract 4 from both sides. This transforms the equation into \(16x = -4.\)
• Finally, divide both sides by 16 to solve for \(x\), giving us \(x = -\frac{4}{16} = -\frac{1}{4}.\)

This approach is methodical and consistent, reflecting the logical path in solving any equation. Isolate, set the condition inside the absolute value, and solve step-by-step.

Intermediate Algebra

Intermediate algebra often deals with more complex concepts than basic algebra. This includes manipulating expressions, equations, and especially working with absolute values.

Key concepts in intermediate algebra involve:

• Understanding the properties of numbers and expressions, such as absolute values and how they affect equations.
• Mastering operations with polynomials, rational expressions, and radicals, similar to how we handle numbers inside absolute values.
• Applying equations solving techniques, not just for straight calculations but in developing logical pathways to solve problems.

Solving absolute value equations, as shown, requires a firm understanding of the rules and applications of intermediate algebra. Through practice, students develop deeper insights into how algebra can model and solve real-world problems.