Problem 46 - Intermediate Algebra

Question

Solve each equation. See Example 3. $$ 15-|12 x+12|=15 $$

Step-by-Step Solution

Answer

The solution is \(x = -1\).

Step 1: Understand the Absolute Value Equation

We start with the equation \( 15 - |12x + 12| = 15 \). To solve this, we first isolate the absolute value by subtracting 15 from both sides, yielding \( -|12x + 12| = 0 \). This implies that \(|12x + 12| = 0\), because subtracting 15 from both sides leaves us with zero.

Step 2: Solve the Absolute Value Equation

The equation \(|12x + 12| = 0\) means that the expression inside the absolute value must equal zero. Hence, we set \(12x + 12 = 0\).

Step 3: Solve for x

Now that we have \(12x + 12 = 0\), we solve for \(x\) by first subtracting 12 from both sides to get \(12x = -12\). Next, divide both sides by 12: \(x = -1\).

Key Concepts

Isolating the Absolute Value

When you face an equation with an absolute value, such as \( 15 - |12x + 12| = 15 \), your first step is to isolate the absolute value expression. Think of this like peeling an onion to get to the core part you need. To do this:

• Eliminate any constants on the same side as the absolute value. Here, subtract 15 from both sides of the equation, which simplifies the equation to \( -|12x + 12| = 0 \).
• Once simplified, you arrive at \(|12x + 12| = 0\). At this point, the absolute value is isolated, meaning the only operation left is the absolute value itself. This isolation step is crucial because it simplifies the problem and prepares you for the next steps.

Remember, absolute value represents the distance a number is from zero on a number line, which makes its initial isolation important in solving such equations.

Setting Expression to Zero

With \(|12x + 12| = 0\) in place, you're tasked with removing the absolute value bars. The solution depends on understanding what the absolute value being zero implies:

• Since absolute value signifies distance from zero and is always non-negative, \(|12x + 12| = 0\) is only true if the expression inside is exactly 0.
• Thus, disregard the absolute value and set the inner expression equal to zero: \(12x + 12 = 0\).
• This transformation leads us directly to a much simpler equation. It's like moving from a complex picture to a straightforward line drawing, where you now focus exclusively on what's inside the absolute value.

This step ensures you're handling the essence of the equation, and thus, positions you perfectly for finding the solution(s) to the variable involved.

Solving for the Variable

Now that you have a simple equation, \(12x + 12 = 0\), it's time to solve for the variable by performing fundamental algebraic operations.

• First, subtract 12 from each side of the equation to get \(12x = -12\). This step is about eliminating any addends attached to the variable.
• Next, divide each side by the coefficient of \(x\), which is 12: \(x = -1\). This operation reveals the value of \(x\) without any other interference.

Now you have the solution, \(x = -1\), which is the value that makes the original equation true. Solving for \(x\) might seem like a puzzle, but breaking it down into these simple steps makes it manageable and even fun!