Problem 22 - Intermediate Algebra

Question

Describe the set graphed below using interval notation. $$ |x-7|=4 $$

Step-by-Step Solution

Answer

The set is \(\{3, 11\}\).

Step 1: Understand Absolute Value Equation

The equation \(|x - 7| = 4\) is an absolute value equation. The expression \(x - 7\) can be either \(4\) or \(-4\) because the absolute value of both numbers is equal to \(4\).

Step 2: Set Up Two Equations

To solve the absolute value equation, you need to set up two separate equations:1. \(x - 7 = 4\)2. \(x - 7 = -4\)

Step 3: Solve the First Equation

Solve the equation \(x - 7 = 4\) by adding \(7\) to both sides:\[x = 4 + 7 = 11\]Thus, one solution is \(x = 11\).

Step 4: Solve the Second Equation

Solve the equation \(x - 7 = -4\) by adding \(7\) to both sides:\[x = -4 + 7 = 3\]Thus, the second solution is \(x = 3\).

Step 5: Write in Interval Notation

Since the solutions to the equation \(|x - 7| = 4\) are \(x = 3\) and \(x = 11\), the number line includes exactly these points. In interval notation, this can be expressed as \(\{3, 11\}\).

Key Concepts

Absolute Value Equations

Absolute value equations might seem complex at first, but they're quite straightforward once you grasp the basics. The absolute value, represented by vertical lines like this: \(|x|\), measures the distance of a number from zero on a number line, without considering direction. So, for any equation

• \(|x - a| = b\)

where \(a\) is a number and \(b\) is a positive value, the expression inside the absolute value bars, \(x - a\), can be either equal to \(b\) or

• \(-b\)

This is because distance is always positive, meaning both \(b\) and \(-b\) have the same absolute value, \(b\). Therefore, when solving, split the equation into two: one for the positive scenario and one for the negative. This will usually give you two potential solutions for \(x\). Understanding these basics about absolute values will simplify the solving process immensely.

Set Notation

Once you solve an equation, it's important to express the solutions in a clear and structured way. This is where set notation comes in handy. Set notation is like listing out the solutions inside brackets. If you have a finite set of numbers as solutions, such as

• \(\{3, 11\}\)

from our example, you simply list them separated by commas inside braces

• \(\{\ldots\}\)

This tells you the complete set of solutions. Set notation is especially useful because of its precision; each solution is distinctly noted, making it easy to understand and visually straightforward. It's crucial to use this method when your solutions are specific numbers rather than ranges.

Solving Equations

Solving equations is a fundamental skill in algebra, and it often comes down to isolating the variable you are solving for. In our example,

• \(x - 7 = 4\)
• and \(x - 7 = -4\)

require simple steps to find \(x\). To solve these, we adjust the equation to get \(x\) by itself. This involves adding or subtracting the same number from both sides to keep the equation balanced.
For instance, add 7 to both sides of the first equation:

• \(x = 4 + 7\)

which leads to \(x = 11\). The same applies to the second equation; adding 7 gives us

• \(x = 3\)

These steps highlight the beauty of algebra: changing numbers to keep equations balanced until we find our solutions. Whether your equation is simple or complex, breaking down the problem into smaller, manageable steps makes solving much easier.