Problem 28 - Intermediate Algebra

Question

Describe the set graphed below using interval notation. $$ |4 x-24.8|=32.4 $$

Step-by-Step Solution

Answer

The set in interval notation is \([-1.9, 14.3]\).

Step 1: Understand the Equation Type

The equation given is an absolute value equation: \(|4x - 24.8| = 32.4\). An absolute value equation \(|A| = B\) means that either \(A = B\) or \(A = -B\). This is because the absolute value of any number is its distance from zero, so it can be either positive or negative.

Step 2: Set Up Two Equations

Based on the properties of absolute values, the equation \(|4x - 24.8| = 32.4\) can be split into two separate equations:1. \(4x - 24.8 = 32.4\) 2. \(4x - 24.8 = -32.4\)

Step 3: Solve the First Equation

Solve \(4x - 24.8 = 32.4\) by first adding 24.8 to both sides to isolate the term with \(x\):\[4x = 32.4 + 24.8\]This simplifies to:\[4x = 57.2\]Next, divide both sides by 4 to solve for \(x\):\[x = \frac{57.2}{4}\]\[x = 14.3\]

Step 4: Solve the Second Equation

Solve \(4x - 24.8 = -32.4\) by first adding 24.8 to both sides:\[4x = -32.4 + 24.8\]This simplifies to:\[4x = -7.6\]Next, divide both sides by 4 to solve for \(x\):\[x = \frac{-7.6}{4}\]\[x = -1.9\]

Step 5: Write Solution in Interval Notation

Now that we have the solutions \(x = 14.3\) and \(x = -1.9\), these represent the two points between which the set is defined. Thus, the interval in which the solutions exist is \([-1.9, 14.3]\). In interval notation, this is written as a closed interval because the solutions include the endpoints \(-1.9\) and \(14.3\).

Key Concepts

Absolute Value Equations

Absolute value equations are a fundamental concept in algebra. They involve expressions within absolute value bars, such as \(|4x - 24.8| = 32.4\). The absolute value refers to the distance a number is from zero on the number line, regardless of direction, making it always non-negative. When solving an equation like \(|A| = B\), we need to understand that there are two possibilities:

• Either the expression inside the absolute value equals \(B\), which can be positive,
• or it equals \(-B\), to cover the negative possibility.

This means we break the absolute value equation into two separate equations. Only by solving both can we find the complete solution set. Grasping this concept helps in tackling a wide array of real-world problems where magnitude, without regard to direction, is considered.

Solving Equations

Solving equations involves finding the values of variables that make the equation true. With absolute value equations, this becomes a little more interesting. Once the absolute value expression is understood and broken down into two cases—as either equal to a positive number or a negative number—you must solve each resulting equation independently. Let's look at our example:

• First, solve the equation \(4x - 24.8 = 32.4\).
• Add 24.8 to both sides to isolate the term with \(x\): \(4x = 57.2\).
• Then divide by 4: \(x = 14.3\).
• Next, solve \(4x - 24.8 = -32.4\) similarly by first adding 24.8 to both sides: \(4x = -7.6\).
• Then, divide by 4 to solve for \(x\): \(x = -1.9\).

By following these logical and arithmetic steps, you can efficiently find the solutions for the absolute value equation. Remember, practice is key to becoming proficient at solving equations!

Interval Solutions

Interval solutions are a way of expressing the range of possible solutions obtained from an equation in a concise manner. After solving an absolute value equation, like \(|4x - 24.8| = 32.4\), you find specific solutions for \(x\) where the expression inside the absolute value equals the given number or its negative counterpart.These solutions define a segment or interval on the number line.

• In our example, we have obtained two solutions: \(x = 14.3\) and \(x = -1.9\).
• These points represent the endpoints of the interval.
Expressing this in interval notation, use brackets \([]\) to denote a closed interval, indicating that the endpoints are included in the solution:
• The interval solution is \([-1.9, 14.3]\).
Using interval notation not only makes it easier to communicate solution sets, but also helps in visualizing the solutions as segments on the number line, perfect for graphical interpretations.