Problem 36 - College Algebra with Modeling and Visualization

Question

Solve the absolute value equation. $$|x-3|=|8-x|$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{11}{2}\).

Step 1: Understand the Absolute Value Equation

The given equation is \(|x-3| = |8-x|\). Absolute value equations can split into two separate cases because the absolute value represents the distance of a number from zero on the number line, regardless of direction.

Step 2: Case 1: Positive Versions are Equal

For the first case, consider both expressions inside the absolute values as positive or equal:\[(x - 3) = (8 - x)\]

Step 3: Solve Case 1 Equation

Solve the equation \((x - 3) = (8 - x)\):1. Add \(x\) to both sides: \[x + x - 3 = 8\] \[2x - 3 = 8\]2. Add 3 to both sides: \[2x = 11\]3. Divide both sides by 2: \[x = \frac{11}{2}\]

Step 4: Case 2: Negate One Expression

For the second case, negate one of the expressions inside the absolute value:\[(x - 3) = -(8 - x)\]

Step 5: Solve Case 2 Equation

Solve the equation \((x - 3) = -(8 - x)\):1. Distribute the negative sign on the right: \[x - 3 = -8 + x\]2. Subtract \(x\) from both sides: \[-3 = -8\]3. Add 8 to both sides: \[5 = 0\], which is a contradiction. Thus, no solution from this case.

Step 6: Verify the Solution from Case 1

Check the solution \(x = \frac{11}{2}\) by substituting it back into the original equation. Both sides must be equal:1. Calculate \(|x-3|\): \[|\frac{11}{2} - 3| = |\frac{5}{2}| = \frac{5}{2}\]2. Calculate \(|8-x|\): \[|8 - \frac{11}{2}| = |\frac{5}{2}| = \frac{5}{2}\]3. Both sides are equal, validating \(x = \frac{11}{2}\).

Key Concepts

Case Analysis in Algebra

In algebra, solving absolute value equations often requires a method called case analysis. This approach involves breaking down the absolute value equation into separate cases. Each case represents a different scenario based on the properties of absolute values. Absolute values measure how far a number is from zero, always yielding a non-negative result.

By analyzing the equation \(|x-3| = |8-x|\), we recognize that it can branch into two cases:

• Both expressions inside the absolute values are non-negative and equal each other: \((x - 3) = (8 - x)\).
• One expression might be the opposite of the other within the absolute value: \((x - 3) = -(8 - x)\).

Each of these cases allows us to consider different scenarios for the values of \(x\), enabling us to solve the equation comprehensively.

This method is crucial when dealing with absolute value equations, ensuring all potential solutions are explored.

Solving Equations

Solving equations is about finding the value or values that make an equation true. In the context of absolute value equations like \(|x-3| = |8-x|\), once we've identified the cases, we can tackle solving each independently.

For the equation displayed in Case 1,

• We equate the two positive expressions: \((x - 3) = (8 - x)\).
• Simplify the equation by combining like terms. Add \(x\) to both sides to start isolating the variable: \(2x - 3 = 8\).
• Next, get rid of the constant by adding 3 to both sides, giving \(2x = 11\).
• Lastly, divide by 2 to solve for \(x\): \(x = \frac{11}{2}\).

In Case 2, where one expression negates the other,

• We set \((x - 3) = -(8 - x)\).
• Distribute the negative sign, resulting in \(x - 3 = -8 + x\).
• Attempting to balance the equation leads to a contradiction: \(5 = 0\), indicating no solutions.

Solving these equations requires careful manipulation of algebraic expressions to isolate the unknown variable and verify any solutions obtained.

Algebraic Expressions

Algebraic expressions form the core of equations like \(|x-3| = |8-x|\). Understanding how to work with these expressions is crucial in algebra. They are combinations of numbers, variables, and operations that convey particular meanings and operations.

In our example, \(x-3\) and \(8-x\) are algebraic expressions. To solve equations involving these, the primary goal is to manipulate the expressions to uncover values of the variable that satisfy the equation. Key operations include adding, subtracting, multiplying, and dividing terms within the expressions.

When managing algebraic expressions with absolute values, remember:

• Identify expressions within the absolute value signs clearly.
• Perform operations within each case separately.
• Be attentive to distributed negative signs and required algebraic operations.

Mastering algebraic expressions leads to a more intuitive understanding of how equations are structured and how they can be solved effectively.