Problem 27 - Algebra 2

Question

Solve each equation. Check your solutions. \(8|w-7|=72\)

Step-by-Step Solution

Answer

The solutions are \(w = 16\) and \(w = -2\).

Step 1: Isolate the Absolute Value

The given equation is \(8|w-7|=72\). First, divide both sides by 8 to isolate the absolute value expression: \[ |w-7| = \frac{72}{8} = 9 \]

Step 2: Set Up Two Equations

The definition of absolute value tells us that if \(|w-7| = 9\), then \(w-7 = 9\) or \(w-7 = -9\). So, we have two equations to solve: 1. \(w-7 = 9\)2. \(w-7 = -9\)

Step 3: Solve the First Equation

Solve the equation \(w-7 = 9\):\[\begin{align*}w - 7 &= 9 \w &= 9 + 7 \w &= 16\end{align*}\]

Step 4: Solve the Second Equation

Solve the equation \(w-7 = -9\):\[\begin{align*}w - 7 &= -9 \w &= -9 + 7 \w &= -2\end{align*}\]

Step 5: Check the Solutions

Substitute \(w = 16\) back into the original equation:\[8|16 - 7| = 8|9| = 72\] This is correct.Substitute \(w = -2\) back into the original equation:\[8|-2 - 7| = 8|-9| = 72\] This is also correct.

Key Concepts

Isolating Absolute Value

To solve an absolute value equation effectively, you must first isolate the absolute value expression. In our exercise, the equation is given as \(8|w-7|=72\). The term \(8|w-7|\) suggests that the expression inside the absolute value bars needs to stand alone.

To achieve this, divide both sides of the equation by 8 to get \(|w-7| = \frac{72}{8} = 9\). This is how the absolute value is isolated successfully.

Once isolated, you can focus on solving the equation by considering both the positive and negative scenarios dictated by the absolute value property.

Solving Linear Equations

With the absolute value isolated, we express the problem in terms of two possible linear equations. Here's why: an absolute value expression \(|x| = a\) implies two scenarios: \(x = a\) and \(x = -a\).

For our example, we set \(w-7 = 9\) and \(w-7 = -9\) and solve each of them separately.

First, deal with the positive case:

• Solve \(w-7 = 9\).
• Start by adding 7 to both sides: \(w = 9 + 7\).
• The solution is \(w = 16\).

Next, the negative case is solved similarly:

• Solve \(w-7 = -9\).
• Add 7 to both sides: \(w = -9 + 7\).
• The solution here is \(w = -2\).

In both scenarios, our linear equations are simplified in just a couple of steps.

Checking Solutions

Checking your solutions is crucial to confirm that they solve the original equation. For each result, substitute back into the original equation to verify.

Let's check \(w = 16\):

• Substitute into the equation: \(8|16 - 7| = 8|9| = 72\).
• This simplifies to 72, agreeing with the original equation.

Now, check \(w = -2\):

• Substitute: \(8|-2 - 7| = 8|-9| = 72\).
• Again, it simplifies to 72, matching the given equation.

Both values are validated, demonstrating their correctness as solutions to the equation \(8|w-7|=72\).