Problem 92 - Algebra 2

Question

Solve each equation. Check your solutions. $$ 5|k-4|=k+8 $$

Step-by-Step Solution

Answer

The solutions are \(k = 7\) and \(k = 2\).

Step 1: Set Up Absolute Value Equations

The given equation is \(5|k-4|=k+8\). Since it's an absolute value equation, consider two cases: \(5(k-4) = k+8\) and \(5(-(k-4)) = k+8\). We'll solve each case separately.

Step 2: Solve the First Case

For the first equation, \(5(k-4) = k+8\), distribute the 5 to get \(5k - 20 = k + 8\). Solve for \(k\) by subtracting \(k\) from both sides: \(4k - 20 = 8\). Add 20 to both sides: \(4k = 28\). Divide by 4: \(k = 7\).

Step 3: Solve the Second Case

For the second equation, \(5(-(k-4)) = k+8\), distribute the 5 to get \(-5(k-4) = k+8\), which simplifies to \(-5k + 20 = k + 8\). Solve for \(k\) by adding \(5k\) to both sides: \(20 = 6k + 8\). Subtract 8 from both sides: \(12 = 6k\). Divide by 6: \(k = 2\).

Step 4: Check Solutions

Substitute \(k = 7\) back into the original equation: \(5|7-4| = 7+8\). This simplifies to \(5(3) = 15\), which equals 15. So, \(k = 7\) is a solution.Substitute \(k = 2\) back into the original equation: \(5|2-4| = 2+8\). This becomes \(5(2) = 10\), which equals 10. So, \(k = 2\) is also a solution.

Key Concepts

Solving Equations

When it comes to solving absolute value equations, the initial step is to remember that absolute values measure the distance a number is from zero. This distance is always non-negative. Hence, when faced with an equation like \(5|k-4| = k+8\), you need to understand that \(|k-4|\) can branch into two scenarios.
The key to solving such equations is by breaking them into two separate linear equations, derived from both the positive and negative interpretations of the absolute value expression:

• The first scenario assumes the expression inside the absolute value is equal to the other side: \(5(k-4) = k+8\).
• The second scenario takes the negative of the expression inside the absolute value: \(5(-(k-4)) = k+8\).

Each case should be solved independently to find possible solutions. This method ensures that all potential solutions are considered, and verifying these in the original equation is crucial to confirm they are valid.

Checking Solutions

Verifying your solutions is a critical part of solving absolute value equations. It helps confirm whether the found values of the variable satisfy the original equation. After determining possible values (like \(k = 7\) and \(k = 2\) in our example), each solution should be plugged back into the initial equation to check its validity.
For instance, substituting \(k = 7\) in the equation \(5|k-4| = k+8\) yields \(5|7-4| = 7+8\), simplifying to \(15 = 15\). Since both sides of the equation are equal, it verifies that \(k = 7\) is indeed a solution.Similarly, substituting \(k = 2\) results in \(5|2-4| = 2+8\), simplifying to \(10 = 10\). This confirms \(k = 2\) is also a valid solution. Checking each potential answer helps avoid overlooking invalid solutions or missing correct ones.

Piecewise Functions

Absolute value functions can be interpreted as piecewise functions. These are functions that have different definitions depending on the interval of the input. This idea is crucial when dealing with equations involving absolute values.
Considering the original equation \(5|k-4| = k+8\), the expression \(|k-4|\) can either represent \(k-4\) or \(-(k-4)\), producing two different scenarios or 'pieces'.

• If \(k-4\) is non-negative (i.e., \(k \geq 4\)), the expression is simply \(k-4\). This leads to the linear equation related to one piece of the function.
• Conversely, if \(k-4\) is negative (i.e., \(k < 4\)), then the expression becomes \(-(k-4)\). This represents the other piece of the function.

This piecewise approach is critical in accurately solving and understanding absolute value problems, as it highlights the diverse possible conditions that the variable \(k\) can satisfy.