Problem 41 - Elementary and Intermediate Algebra

Question

Solve each equation. \(2|3 x+24|=0\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Absolute Value Equation

The original equation is \(2|3x + 24| = 0\). This equation states that twice the absolute value of the expression \(3x + 24\) equals 0. Since an absolute value is always non-negative, the only way this equation holds true is if the expression inside the absolute value itself is equal to zero.

Step 2: Set the Expression Inside the Absolute Value to Zero

Since the only possibility for an absolute value to be 0 is for its argument to be 0, we set the expression inside the absolute values equal to zero. That is: \[ 3x + 24 = 0 \]

Step 3: Solve for x

We now solve the equation \(3x + 24 = 0\) to find the value of \(x\). Start by subtracting 24 from both sides:\[ 3x = -24 \]Then, divide both sides by 3 to isolate \(x\):\[ x = -8 \]

Key Concepts

Solving Equations

Solving equations is like unraveling a math mystery by finding the value of the unknown variable, typically represented by letters like \(x\). The ultimate goal is to find out what value of \(x\) makes the equation true. For our given equation, the unknown variable \(x\) is hidden inside an absolute value expression. We must follow specific steps to solve it.

• Start by simplifying the equation. That often involves combining like terms or expanding expressions on each side.
• If there is an absolute value, decide when it equals zero, as absolute values are always non-negative.
• Rework the simplified expression to further isolate the variable.

In the exercise we have, simplifying means turning \(2|3x + 24| = 0\) into \(3x + 24 = 0\) by recognizing that twice an absolute zero is still zero.

Algebra

Algebra is the branch of mathematics where we represent numbers with symbols, making it easier to solve various types of equations. In our example, the equation involves an expression formatted around variables and constants. How do we solve it using algebraic principles?

• Expression: Composed of parts like \(3x + 24\), including both variables and constants.
• Operations: Using arithmetic operations such as addition, subtraction, multiplication, and division carefully helps us isolate the variable.
• Balancing: Perform operations equally on both sides of the equation to keep it balanced and valid.

For instance, subtracting 24 from both sides of the equation \(3x + 24 = 0\) is a common algebraic step to balance and simplify, leading us to \(3x = -24\). Consistently applying algebraic rules helps solve equations methodically.

Absolute Value Properties

The concept of absolute value can be tricky, but understanding it is essential for solving specific equations. The absolute value of a number represents its distance from zero on a number line, displaying only as positive or zero.

• Absolute Zero: The only value for which an absolute value of any expression results in zero is when the expression itself is zero. This principle guided us to set \(3x + 24 = 0\).
• Simplify: Always remove or eliminate the absolute value by solving for its contained expression first.
• Non-Negative Nature: Since absolute values can't be negative, when they are multiplied to form zero as in \(2|3x + 24| = 0\), the expression inside should also be zero.

These properties enable us to simplify and resolve the original equation, echoing the steps above of setting the expression equal to zero and solving for \(x = -8\). Understanding these properties allows easier manipulation and resolution of equations involving absolute values.