Problem 45 - Elementary and Intermediate Algebra

Question

Solve each equation. \(6-3|10 x+5|=6\)

Step-by-Step Solution

Answer

The solution is \(x = -\frac{1}{2}\).

Step 1: Simplify the Equation

First, simplify the equation by isolating the absolute value expression. Begin by subtracting 6 from both sides: \[ 6 - 3|10x + 5| = 6 \]Subtract 6 from both sides to get: \[-3|10x + 5| = 0\]

Step 2: Isolate the Absolute Value

Since the coefficient of the absolute value is \(-3\), divide both sides by \(-3\) to completely isolate the absolute value term: \[|10x + 5| = 0\]

Step 3: Solve for x

Now that the absolute value is isolated and equal to 0, the expression inside the absolute value must also be 0. Set up the equation: \[10x + 5 = 0\]Solve for \(x\) by subtracting 5 from both sides: \[10x = -5\]Then divide by 10: \[x = -\frac{1}{2}\]

Key Concepts

Exploring Absolute Value Equations

Absolute value equations often confuse students, but understanding them can become much simpler with practice. An absolute value, represented as \(|x|\), is the distance of a number \(x\) from zero on the number line. It's always non-negative, as it measures a distance. In absolute value equations, we are often tasked with finding what number or numbers make the expression inside the absolute value equal to the given value.
In the equation \(|10x + 5| = 0\), we focus first on what's inside the absolute value. Since absolute value describes distance, it equals zero only at its origin. This means that the expression inside must also be equal to zero. Therefore, solving what's inside the absolute expression becomes our main task.
Understanding absolute value as a concept of distance helps us interpret and solve these problems systematically. It's just like finding the point where two locations on a map meet at zero distance in between.

Mastering Algebraic Manipulation

Algebraic manipulation is the toolset that helps us solve mathematical equations and expressions. It includes operations like addition, subtraction, multiplication, division, and factoring, which we use to isolate variables or simplify expressions.
In our exercise, we started with the equation \(6 - 3|10x + 5| = 6\). Here, our first move involves simplifying by subtracting 6 from both sides, resulting in \(-3|10x + 5| = 0\). This action clears away extra terms, allowing us to focus on isolating \(|10x + 5|\).
Once the absolute value expression is isolated, we divide by \(-3\). This step ensures that the coefficient of the absolute value is 1, making it easier to solve the subsequent equation. The careful arrangement and manipulation of terms lay a foundation for accurately finding solutions to algebraic problems.

Approaching Mathematical Problem Solving

Mathematical problem solving involves a strategic approach to tackle equations, based on logical reasoning and methodical planning. Each problem may require different techniques, but a systematic process can be applied across various kinds of equations.
To solve \(6-3|10 x+5|=6\), we break the problem into smaller, manageable steps. Initially, we isolate the absolute value term. With \(-3|10x + 5| = 0\), dividing both sides by \(-3\) helps in simplifying the problem to the basic absolute value equation \(|10x + 5| = 0\).
Our final task is to resolve the remaining equation \(10x + 5 = 0\). This involves logical deductions, such as subtracting 5 from both sides, and then dividing by 10. In the end, we solve for \(x\), discovering that \(x = -\frac{1}{2}\). Approach every problem with a clear plan, and always check your steps to ensure accuracy.