Problem 74 - College Algebra

Question

Absolute value expressions are equal when the expressions inside the absolute value bars are equal to or opposites of each other. $$ |2 x-7|=|x+3| $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 10\) and \(x = 4/3\)

Step 1: Set up two equations

Start by setting up two separate equations according to what was mentioned about the absolute value expressions being equal or opposites. These two equations are \(2x - 7 = x + 3\) and \(2x - 7 = -(x + 3)\)

Step 2: Solve the first equation for x

To solve \(2x - 7 = x +3\), first subtract x from both sides to get \(x - 7 = 3\). Then, add 7 to both sides to get the solution \(x = 10\)

Step 3: Solve the second equation for x

To solve \(2x - 7 = -(x + 3)\), first distribute the negative sign on the right side of the equation to get \(2x - 7 = -x - 3\). Then, add x to both sides to get \(3x - 7 = - 3\). Lastly, add 7 to both sides to get the solution \(x = 4/3\)

Key Concepts

Solving Absolute Value Equations

When we come across an absolute value equation, such as \(|2x - 7| = |x + 3|\), we're tasked with finding the x-values that make the equation true. The absolute value bars indicate the distance of a number from zero on the number line, regardless of direction.

To solve these, we essentially consider two separate scenarios: one where the expressions inside the bars are equal without changing the sign, and one where they are equal when one expression is the opposite in sign. In our case, the two resulting equations are \(2x - 7 = x + 3\) and \(2x - 7 = -(x + 3)\). This offers a structured approach to breaking down what may at first seem like a more complex problem.

By solving each of these linear equations separately and checking that the solutions satisfy the original absolute value equation, we ensure a thorough and accurate answer, accounting for the fact that absolute value expressions can be equal to both positive and negative versions of a number.

Setting Up Equations

The foundation of solving absolute value equations is properly setting up the equations to be solved. Starting with an equation like \(|2x - 7| = |x + 3|\), we use the understanding that expressions within absolute value bars can be either equal or opposites.

For teaching purposes, it's vital to convey the idea that when we see absolute value bars, we're dealing with two potential cases – one where the value inside is not affected by the bars (positive) and one where we must consider the value as negative. This dual-nature of absolute values is crucial for students to grasp, and it is a concept that extends well into more advanced mathematics.

By emphasizing these cases and demonstrating how to translate an absolute value equation into a pair of standard linear equations, we empower students to tackle a broader range of problems with confidence.

• If the expressions are equal: \(2x - 7 = x + 3\)
• If they are opposites: \(2x - 7 = -(x + 3)\)

Once set correctly, solving these equations is just a matter of applying basic algebraic principles.

Algebraic Manipulation

Algebraic manipulation is the bread and butter of solving any algebraic equation, including those involving absolute values. Once our equations are established—\(2x - 7 = x + 3\) and \(2x - 7 = -(x + 3)\)—we can start manipulating them to isolate the variable x.

The first equation simplifies to \(x = 10\) after subtracting x from both sides and then adding 7. In the second equation, distributing the negative sign and adding x to both sides gives us \(3x - 7 = -3\), which resolves to \(x = \frac{4}{3}\). These algebraic steps are standard: combining like terms, distributing multiplication over addition or subtraction, and using inverse operations to isolate the variable.

Understanding these manipulative techniques is essential for students not only to solve the problem at hand but also to develop the skillset required for more complex algebra problems. Clear explanation of each step can demystify the process and anchor the student's learning experience in solid, comprehensible methods.