Problem 41 - Intermediate Algebra

Question

Solve each equation. See Example 3. $$ 2|3 x+24|=0 $$

Step-by-Step Solution

Answer

The solution is \(x = -8\).

Step 1: Understand the Absolute Value Equation

The given equation is \(2|3x+24|=0\). Absolute value equations are solved by isolating the absolute value expression \(|3x+24|\) and considering the case when it equals to zero since it's multiplied by 2 which already equals zero.

Step 2: Simplify the Equation

Since \(2|3x+24|=0\), we can divide both sides by 2 to simplify: \(|3x+24| = 0\).

Step 3: Solve for the Expression Inside the Absolute Value

For \(|3x+24| = 0\), the expression inside the absolute value, \(3x + 24\), must be equal to zero as well. Hence, we set up the equation \(3x + 24 = 0\).

Step 4: Solve the Linear Equation

To solve \(3x + 24 = 0\), subtract 24 from both sides to get \(3x = -24\). Then divide by 3: \(x = -8\).

Step 5: Verify the Solution

Substitute \(x = -8\) back into the original expression to ensure correctness: \(|3(-8) + 24| = | -24 + 24 | = |0| = 0\), which satisfies the equation.

Key Concepts

Linear Equations

Linear equations are a fundamental part of algebra, involving functions of the first degree. In these equations, each term is either a constant or a product of a constant and the variable. For example, in the equation \(3x + 24 = 0\), 3 is the coefficient of the variable \(x\), and 24 is a constant term.
Solving a linear equation involves finding the value of the variable that makes the equation true. Here’s a simple process to follow:

• Isolate the variable by using inverse operations to cancel out other terms.
• Perform the same operation on both sides of the equation to maintain equality.
• Simplify the expression until you solve for the variable.

In the exercise, we find the solution by first subtracting 24 and then dividing by 3, leading us to \(x = -8\). This simple, systematic approach is what makes solving linear equations straightforward and reliable.

Absolute Value

Absolute value refers to the "distance" of a number from zero on the number line, regardless of direction. It is always non-negative. For example, the absolute value of both 5 and -5 is 5, represented as \(|5| = 5\) and \(|-5| = 5\).
In equations, absolute values have special properties. An equation like \(|3x+24| = 0\) can seem a bit confusing due to the presence of the absolute value operator. However, understanding that absolute value can only be zero if the argument (what's inside) is zero makes it simpler.
In this exercise, \(|3x+24| = 0\), we see that the task is to isolate \(3x+24\) and solve it directly for zero. This step reveals the importance of understanding what absolute value really signifies when solving equations.

Equation Solving Techniques

When solving equations, several strategies are beneficial. These techniques help in breaking down complex equations into more manageable parts.

• Isolation: Always aim to isolate the unknown variable. This helps in simplifying the equation significantly.
• Understanding Operation: Recognize operations involved and perform inverse operations to simplify. For instance, addition involves subtraction and multiplication involves division.
• Verification: After finding a solution, substitute it back into the original equation to ensure correctness. This eliminates the possibility of extraneous solutions.

In our initial equation, \(2|3x+24|=0\), the first step was to divide both sides by 2. After isolating the absolute value, we solved \(3x+24=0\) for the variable \(x\). These techniques are essential for successful equation solving, leading to accurate results and better understanding.