Problem 74 - Algebra 2

Question

Solve each equation. Check your solutions. \(10=2|w+6|\)

Step-by-Step Solution

Answer

The solutions are \(w = -1\) and \(w = -11\).

Step 1: Isolate the Absolute Value

Start with the given equation: \(10 = 2|w+6|\). To isolate the absolute value, divide both sides by 2 to get \(|w+6| = 5\).

Step 2: Remove Absolute Value

Since \(|w+6| = 5\), this implies two possible equations: \(w+6 = 5\) and \(w+6 = -5\).

Step 3: Solve First Equation

Solve the equation \(w+6 = 5\). Subtract 6 from both sides to find \(w = -1\).

Step 4: Solve Second Equation

Solve the equation \(w+6 = -5\). Subtract 6 from both sides to find \(w = -11\).

Step 5: Check the Solutions

Substitute \(w = -1\) back into the original equation: \(2|-1+6| = 2|5| = 10\), which is true. Substitute \(w = -11\) into the original equation: \(2|-11+6| = 2| -5| = 10\), which is also true. Thus, both solutions are valid.

Key Concepts

Isolating Absolute Value

The first step in solving an absolute value equation is to isolate the absolute value expression. In the problem we're discussing, we start with the equation \(10 = 2|w+6|\).
To easily manage and solve the equation, we need to make sure that the absolute value is by itself on one side.

• Begin by dividing both sides of the equation by the number multiplying the absolute value (in this case, 2).
• This simplification leads us to \(|w+6| = 5\).

Isolating the absolute value helps to set up the next steps of solving, as it simplifies the equation to understand the next possible values \(w\) can take. It turns the focus to what is inside the absolute value.

Removing the Absolute Value and Solving

Once the absolute value is isolated in the equation \(|w+6| = 5\), the goal is to eliminate the absolute value. Absolute values measure the distance from zero, so \(|x| = a\) results in two potential scenarios:

• \(x = a\)
• \(x = -a\)

This means we set up two separate linear equations: \(w+6 = 5\) and \(w+6 = -5\). Solving these equations independently:- For \(w+6 = 5\): Subtract 6 from both sides to find that \(w = -1\).- For \(w+6 = -5\): Subtract 6 again to solve \(w = -11\).This dual approach covers all numbers that could have an absolute value of 5, thereby ensuring all potential solutions are found. It is essential to solve both equations to capture both potential values that solve the original absolute value equation.

Checking Solutions to Absolute Value Equations

After finding solutions to the absolute value equation, the final and crucial step is to verify them. This ensures that the solutions indeed satisfy the original equation: \(10 = 2|w+6|\). - **For** \(w = -1\): Plugging back into the original equation, we calculate: \[2|-1+6| = 2|5| = 2(5) = 10\]The left side equals the right side, confirming that \(w = -1\) is a correct solution.- **For** \(w = -11\): Similarly, substitute into the equation:\[2|-11+6| = 2|-5| = 2(5) = 10\]Again, the equation holds true, validating \(w = -11\) as well.Checking solutions is a critical step to catch any potential mistakes or miscalculations. Always substitute your solutions back into the original equation to ensure that they are correct. This step gives confidence in your results and helps solidify your understanding of solving these types of equations.