Problem 84 - Intermediate Algebra

Question

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. $$ |8-x|=|x+2| $$

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Understand Absolute Value Equations

When dealing with absolute value equations such as \(|a| = |b|\), the equation can lead to two separate cases: either \(a = b\) or \(a = -b\). For this particular problem, this means we should set up two equations: \(8-x = x+2\) and \(8-x = -(x+2)\).

Step 2: Solve the First Case

Let's solve the equation from Case 1: \(8-x = x+2\). Add \(x\) to both sides to get \(8 = 2x + 2\). Then subtract 2 from both sides to yield \(6 = 2x\). Finally, divide both sides by 2 to solve for \(x\), resulting in \(x = 3\).

Step 3: Solve the Second Case

Now, solve the equation from Case 2: \(8-x = -(x+2)\). This simplifies to \(8-x = -x - 2\). Add \(x\) to both sides to get \(8 = -2\), then add 2 to both sides to find \(10 = -2x\). Finally, divide both sides by \(-2\) to solve for \(x\), giving \(x = -5\).

Step 4: Confirm the Solutions

To verify, substitute \(x = 3\) and \(x = -5\) back into the original equation \(|8-x| = |x+2|\). For \(x = 3\), the equation becomes \(|8-3| = |3+2|\) or \(|5| = |5|\), which holds true. For \(x = -5\), it becomes \(|8 - (-5)| = |-5 + 2|\) or \(|13| = |3|\), and this does not hold. Thus, \(x = -5\) is not a valid solution.

Step 5: Final Answer and Graphing

The only valid solution for the equation is \(x = 3\). Inequalities do not need graphing here as it is a simple equation.

Key Concepts

Solving Equations

Solving absolute value equations involves understanding the fundamental concept that absolute values measure a number's distance from zero on a number line. For equations like \(|a| = |b|\), this distance can be equated to either \(a = b\) or \(a = -b\). These two scenarios give us different equations to solve.

In the provided original solution, two separate equations are examined:

• \(8-x = x+2\)
• \(8-x = -(x+2)\)

Each equation deals with a different case of the absolute value, ensuring that all possible solutions are found. By solving these equations sequentially, we simplify each expression step-by-step until the variable is isolated. This stepwise approach is similar in many algebraic solutions, where the goal is to make sense of a potentially complex problem by breaking it down.

In this example, solving leads to possible solutions for \(x\) being 3 or -5. However, further validation shows only \(x = 3\) is correct when substituted back into the original equation to check for consistency.

Algebraic Expressions

Algebraic expressions form the building blocks of equations and inequalities. They are composed of constants, variables, and operations. In absolute value equations such as \(|8-x|=|x+2|\), we encounter expressions on both sides that must be handled with care.

Understanding how to manipulate and rearrange these expressions is crucial. Here, expressions such as \(8-x\) and \(x+2\) involve both addition and subtraction operations along with the variable \(x\). During the process:

• Terms are rearranged.
• Operations are applied, such as adding \(x\) to both sides to simplify.
• Dividing both sides of an equation by the same non-zero number to solve for \(x\).

These manipulations follow the rules of algebra and ensure the equation remains balanced, leading towards finding the variable's value that satisfies the equation. As you solve more equations, understanding these basic operations and their application becomes a valuable skill for tackling more complex problems.

Interval Notation

Interval notation provides a compact way to express a set of solutions, especially for inequalities. Though our original exercise involved an equation, interval notation often comes into play when solutions are more than just single points.

In interval notation, solutions of inequalities are represented as intervals, describing where solutions lie on a number line. This method is especially useful for showcasing solution sets where there are ranges or multiple values:

• A single solution, such as \(x = 3\), doesn't require interval notation, but could simply be expressed as \([3]\).
• For ranges, if, for instance, \(x\) were to satisfy \(3 < x < 7\), we'd use interval notation like \((3, 7)\).

This notation helps in simplifying complex solution sets visually and analytically, providing clarity on which values are included or excluded. While the current exercise focused mainly on solving an equation, gaining a thorough understanding of interval notation will aid future endeavors, particularly when tackling inequalities or equations with broader solution sets.