Problem 45 - Intermediate Algebra

Question

Solve each equation. See Example 3. $$ 6-3|10 x+5|=6 $$

Step-by-Step Solution

Answer

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

Step 1: Isolate the Absolute Value

To begin solving the equation \(6 - 3|10x + 5| = 6\), first isolate the absolute value term by subtracting 6 from both sides to get \(-3|10x + 5| = 0\).

Step 2: Remove Negative Coefficient

Simplify the equation \(-3|10x + 5| = 0\) by dividing both sides by -3, yielding \(|10x + 5| = 0\).

Step 3: Solve the Absolute Value Equation

The equation \(|10x + 5| = 0\) implies that the expression inside the absolute value sign must be zero, thus \(10x + 5 = 0\).

Step 4: Solve for x

Subtract 5 from both sides of the equation \(10x + 5 = 0\) to get \(10x = -5\). Then, divide both sides by 10 to solve for \(x\), resulting in \(x = -\frac{1}{2}\).

Key Concepts

Isolating Absolute Values

When working with absolute value equations, isolating the absolute value is often your first task. The absolute value sign, denoted as \(|x|\), transforms any number inside it to its non-negative form. To solve an equation with an absolute value, you have to first ensure that this absolute value term stands alone on one side of the equation.
For example, consider the equation \(6 - 3|10x + 5| = 6\). We need to isolate \(|10x + 5|\) for easier handling. Start by subtracting 6 from both sides, which cancels out the 6 on the left, resulting in \(-3|10x + 5| = 0\).
This subtraction is crucial as it allows us to focus solely on the absolute value term, making further steps much simpler. Remember, isolating absolute values can involve addition, subtraction, or division, depending on your specific equation. The main goal is to have just the absolute value on one side, free from other numbers or variables.

Solving Linear Equations

Once you've successfully isolated the absolute value term, you're likely to encounter a linear equation needing more straightforward solving techniques. A linear equation is any equation that can be expressed in the form \(ax + b = c\).
In our example, after isolating the absolute value, we were left with the equation \(|10x + 5| = 0\). To progress further, recognize that the absolute value of anything equals zero only if the expression inside it also equals zero. Thus, you can rewrite the equation without the absolute value bars: \(10x + 5 = 0\).
This form, \(ax + b = 0\), is a typical linear equation. Linear equations are simple to handle; they require basic operations such as addition, subtraction, multiplication, or division to solve for \(x\). Such operations keep the equation balanced while isolating the variable.

Steps to Solve Equations

Solving equations is an organized process, much like following a recipe. It helps to break down the task into smaller, easily manageable steps. Here’s how to approach it:
1. **Isolate Absolute Values:** Begin with isolating any absolute value terms, as discussed earlier. This prepares the equation for more direct solution methods.
2. **Remove Coefficients:** Get rid of any coefficients attached to the absolute value by performing allowable operations like division. For example, divide both sides by any occurances of negative coefficients to simplify the expression. In our situation, dividing \(-3|10x + 5| = 0\) by \(-3\) gave us \(|10x + 5| = 0\).
3. **Solve the Absolute Value Equation:** Convert the absolute value equation into a linear equation, if possible, by setting the inside of the absolute value equal to zero.
4. **Solve for Variable:** Finally, solve for the variable using conventional linear equation techniques. This involves simplifying the equation through basic operations until you isolate the variable completely on one side.
Following these steps can significantly streamline the solving process, reduce errors, and improve your confidence in handling equations accurately.