Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-14,\) write the solution set of each equation. $$ |2 x-5|+2=13 $$

Step-by-Step Solution

Answer

The solution set is \( \\{-3, 8\\} \).

Step 1: Isolate the Absolute Value Expression

First, we need to isolate the absolute value expression in the equation. We start with the equation: \[|2x - 5| + 2 = 13\]Subtract 2 from both sides of the equation to isolate the absolute value term:\[|2x - 5| = 11\]

Step 2: Set Up Two Separate Equations

Since we have an absolute value equation, we know that the expression inside the absolute value can be either positive or negative. This gives us two separate equations to solve:1. \(2x - 5 = 11\)2. \(2x - 5 = -11\)

Step 3: Solve First Equation

Solve the first equation, \(2x - 5 = 11\):Add 5 to both sides:\[2x = 16\]Then divide both sides by 2:\[x = 8\]

Step 4: Solve Second Equation

Solve the second equation, \(2x - 5 = -11\):Add 5 to both sides:\[2x = -6\]Then divide both sides by 2:\[x = -3\]

Step 5: Write the Solution Set

We now have two solutions for the equation \(|2x - 5| + 2 = 13\). Thus, the solution set is:\[x = \{-3, 8\}\]

Key Concepts

Solution Set

In the realm of algebra, the "solution set" of an equation is incredibly important. It's essentially the collection of all possible values that will satisfy an equation, making it true. For the given absolute value equation \(|2x - 5| + 2 = 13\), we are tasked to find the specific values for \(x\) that will work. After working through the equation, we found the solution set to be \(\{-3, 8\}\). This tells us that if \(x\) is either \(-3\) or \(+8\), the original equation will hold true. Always remember, when dealing with equations, determining the solution set is the final step to conclude your solution.

Isolate Absolute Value

The first crucial step in solving any absolute value equation is to isolate the absolute value expression. In our example, this means we start with the equation: \[ |2x - 5| + 2 = 13 \] We need to remove any additional elements from around the absolute value to deal with it directly. For this reason, we subtract 2 from both sides, leading to: \[ |2x - 5| = 11 \] This step is important because it allows us to focus solely on the expression within the absolute value, making it easier to take our next steps in solving the equation. Essentially, isolation prepares the equation for further manipulation.

Two Separate Equations

When working with absolute value equations, a key insight is to recognize that the expression inside can represent two cases: one positive, and one negative. This is why we set up two separate equations. For the isolated equation \(|2x - 5| = 11\), this results in:

• \(2x - 5 = 11\)
• \(2x - 5 = -11\)

By breaking the problem into two conditions, we thoroughly explore every possible way the original expression fulfills the equation. This step greatly expands our ability to find every potential solution and is a foundational method in algebra for solving absolute value expressions.

Solving Linear Equations

Once we have our two separate linear equations, we solve them independently. Let's walk through this:

• First Equation: \(2x - 5 = 11\)
• Add 5 to both sides to get: \(2x = 16\)
• Divide by 2 to solve for \(x\): \(x = 8\)

• Second Equation: \(2x - 5 = -11\)
• Add 5 to both sides to get: \(2x = -6\)
• Divide by 2 to solve for \(x\): \(x = -3\)

Solving these linear equations allows us to find the values of \(x\) that make each equation true. Therefore, with both solutions in hand, we finalize our solution set as \(\{-3, 8\}\). Remember, solving linear equations is straightforward yet essential in concluding the results of our absolute value equations.