Problem 13 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$\left|\frac{x-4}{2}\right|=5$$

Step-by-Step Solution

Answer

The solutions are \(x = 14\) and \(x = -6\).

Step 1: Set Up Two Equations

An absolute value equation \(\bigg|\frac{x-4}{2}\bigg| = 5\) means the expression inside the absolute value can be equal to 5 or -5. So first, set up two separate equations: \[\frac{x-4}{2} = 5\] and \[\frac{x-4}{2} = -5\].

Step 2: Solve the First Equation

Take the first equation \[\frac{x-4}{2} = 5\]. Multiply both sides by 2 to get rid of the fraction: \[x-4 = 10\]. Then, add 4 to both sides to isolate x: \[x = 14\].

Step 3: Solve the Second Equation

Next, solve the second equation \[\frac{x-4}{2} = -5\]. Again, multiply both sides by 2: \[x-4 = -10\]. Then, add 4 to both sides: \[x = -6\].

Step 4: Check the Solutions

Check both solutions in the original equation to verify correctness. For \[x = 14\]: \(\bigg|\frac{14-4}{2}\bigg| = \bigg|\frac{10}{2}\bigg| = 5\), which is correct. For \[x = -6\]: \(\bigg|\frac{-6-4}{2}\bigg| = \bigg|\frac{-10}{2}\bigg| = 5\), which is also correct.

Key Concepts

Equation Solving

Solving equations is a key aspect of algebra that you will use in many areas of math and science. In this problem, you are given an absolute value equation: \(\bigg|\frac{x-4}{2}\bigg|=5\). The absolute value function measures the distance of a number from zero on the number line. This means what's inside the absolute value can be both positive and negative.
First, you will convert this single absolute value equation into two separate equations: \[\frac{x-4}{2}=5\] and \[\frac{x-4}{2}=-5\].
Solving these involves standard algebraic steps like removing the fraction by multiplying both sides by 2, and isolating \[x\] by adding or subtracting values. Breaking the equation down into these manageable steps will help you find both possible solutions.
Always remember with absolute value equations, you get two cases – one for the positive scenario and one for the negative scenario. Once solved, check both solutions in the original equation to ensure they are correct.

Absolute Value

The absolute value of a number represents its distance from zero on the number line. It is always non-negative. For example, the absolute value of both +5 and -5 is 5, which is written as: \[|5| = 5\] and \[|-5| = 5\].
In our exercise, we encounter an absolute value equation: \[\bigg|\frac{x-4}{2}\bigg|=5\]. This expresses that the distance from zero of whatever is inside the absolute value symbols equals 5. This is why we set up two equations: \[\frac{x-4}{2}=5\] and \[\frac{x-4}{2}=-5\]. This ensures we consider both the positive and negative scenarios that could solve the absolute value equation.
When solving absolute value equations, always check both solutions in the original absolute value equation. This is a crucial step to verify their correctness.

Precalculus

Precalculus involves the study of functions, equations, and analytical geometry that prepare you for calculus. Understanding absolute values and how to solve related equations is fundamental at this level.
In our problem, the absolute value equation \[\bigg|\frac{x-4}{2} \bigg|=5\] helps build skills for handling more complex equations and functions. Learning to transform an absolute value equation into two linear equations is a valuable skill. This transformation simplifies solving the problem and lays a foundation for understanding piecewise functions.
Practice regularly with different types of absolute value equations to strengthen your problem-solving skills. Remember, each equation may require different approaches, but the logic of isolating the variable remains consistent across various problems. Understanding these concepts thoroughly will provide a strong foundation for tackling higher-level precalculus and calculus topics.