Problem 61 - College Algebra

Question

Solve each absolute value equation or indicate the equation has no solution. $$ |x-2|=7 $$

Step-by-Step Solution

Answer

The solutions for the equation are \( x = 9 \) and \( x = -5 \).

Step 1: Investigate the first scenario (x - 2 = 7)

Start solving the equation considering positive \( x \). Here, \( x - 2 \) is equal to 7. That being, an algebraic manipulation results in the value of \( x = 7 + 2 \), which gives \( x = 9 \).

Step 2: Investigate the second case (x - 2 = -7)

Now, consider \( x - 2 \) is equal to -7. Manipulating the equation algebraically, the value of \( x = -7 + 2 \), which solves to \( x = -5 \).

Step 3: Summarize the solutions

Having taken into account the two possible cases given the properties of absolute values, it's concluded that the solutions for the equation \( |x-2|=7 \) are \( x = 9 \) and \( x = -5 \).

Key Concepts

Algebraic Manipulation

When solving absolute value equations like \( |x-2|=7 \), algebraic manipulation is a valuable tool. This process involves rearranging and simplifying parts of the equation to isolate the variable of interest—in our case, \( x \). To effectively apply algebraic manipulation, one must understand the order of operations and the balance between both sides of the equation.

• Consider \( |x-2| \) like any other algebraic expression. Initially, ignore the absolute value and solve the equation as though it were \( x-2=7 \). This gives us one possible value of \( x \) after moving the 2 to the other side of the equation.
• The second step is to acknowledge the unique principle of absolute values, which can also be negative and still satisfy the equation. So, we solve \( x-2=-7 \) as well, providing us with another valid solution for \( x \).

Through these steps, algebraic manipulation leads us to two possible solutions for the given absolute value equation. It’s important to check that each solution, when plugged back into the original equation, makes the equation true.

Solution of Equations

The process of finding the solution of equations, especially absolute value equations, requires a careful approach that deals with both the algebraic structure and the specific properties of the functions involved.

For an absolute value equation like \( |x-2|=7 \), the solution is not immediately apparent. Here’s how we approach it:

• Separate the equation into two scenarios based on the definition of absolute value. This yields the equations \( x-2=7 \) and \( x-2=-7 \).
• Solve each resulting equation. The solution to the first is \( x=9 \) and the second is \( x=-5 \).
• Verifying each solution by substituting it back into the original equation will confirm that both solutions are correct.

This method ensures we find all potential solutions to the equation. In cases where no solutions exist, the simpler verification step would showcase the contradiction.

Properties of Absolute Values

Understanding the properties of absolute values is crucial in solving equations that include absolute value expressions. An absolute value measures the distance between a number and zero on the number line, regardless of direction. This property can lead to two different mathematical scenarios, hence why the solution to an absolute value equation can result in two numerical values.

Here's an illustration:

• For the equation \( |x-2|=7 \), \( x-2 \) must be 7 units away from zero. It can be 7 units in the positive direction (\( x-2=7 \)) or 7 units in the negative direction (\( x-2=-7 \)).
• These two scenarios correspond to the two possible values of \( x \), demonstrating that \( x \) can exist at two different points on the number line that are equidistant from 2.

The properties of absolute values necessitate that we explore all potential solutions by considering both the positive and negative outcomes. Only through this comprehensive approach can the full set of solutions be revealed.