Problem 32 - Algebra A Combined Function

Question

Solve each inequality. Write the solution set in interval notation. $$ \frac{x+10}{x-10}>0 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, -10) \cup (10, \infty)\).

Step 1: Identify the critical points

To solve the inequality \( \frac{x+10}{x-10}>0 \), you need to identify the points where the fraction could be zero or undefined. This occurs at \( x+10=0 \) and \( x-10=0 \). Solving these equations gives critical points: \( x=-10 \) and \( x=10 \).

Step 2: Determine the test intervals

The critical points \( x=-10 \) and \( x=10 \) divide the number line into three intervals: \((-\infty, -10)\), \((-10, 10)\), and \((10, \infty)\).

Step 3: Test each interval

For each interval, pick a test point and substitute it into the inequality to determine if the interval satisfies the inequality. - For \((-\infty, -10)\), choose \( x = -11 \): \[ \frac{-11+10}{-11-10} = \frac{-1}{-21} > 0 \]- For \((-10, 10)\), choose \( x = 0 \): \[ \frac{0+10}{0-10} = \frac{10}{-10} < 0 \]- For \((10, \infty)\), choose \( x = 11 \): \[ \frac{11+10}{11-10} = \frac{21}{1} > 0 \]

Step 4: Write the solution in interval notation

The solution consists of the intervals where the inequality \( \frac{x+10}{x-10}>0 \) holds true: therefore, the solution is the union of intervals \((-\infty, -10)\) and \((10, \infty)\). These are the intervals where the expression is greater than zero.

Key Concepts

Understanding Critical Points

In the context of inequalities, critical points are key values that help us understand where changes occur on a graph. For our inequality \(\frac{x+10}{x-10}>0\), these critical points are identified by setting the numerator and denominator equal to zero. This is because these points can indicate where the expression crosses zero or becomes undefined.
Here, you solve the equations \(x+10=0\) and \(x-10=0\) to find the critical points. The solutions are \(x=-10\) and \(x=10\). These values split the number line into distinct intervals, helping you test where the inequality holds true or false.

• \(x=-10\): The point where the expression \(x+10\) equals zero.
• \(x=10\): The point where the expression \(x-10\) equals zero, making the denominator zero, hence undefined.

Critical points are crucial in solving inequalities because they help outline the regions you need to evaluate when determining the solution set.

Interval Notation Demystified

Interval notation is a concise way of expressing a range of values in mathematics. When using interval notation, we describe all the numbers between two values on the number line. This is particularly useful when representing the solution sets of inequalities, like in our problem.
In the given inequality problem, once we have our critical points, we divide the number line into intervals. These intervals represent different ranges of values between and outside of these points:

• \(( -\infty, -10)\)
• \((-10, 10)\)
• \((10, \infty)\)

Each interval is tested with a chosen point from within, to see if it satisfies the inequality. Interval notation makes it easy to express the union of intervals, which combines all possible solutions. For example, the solution to our problem is written using interval notation as \((-\infty, -10) \cup (10, \infty)\), indicating where the expression is greater than zero.

Defining the Solution Set

Once we have determined the intervals where an inequality holds true, we can define the solution set of an inequality. The solution set is simply the collection of all x-values that satisfy the inequality condition. In our inequality \(\frac{x+10}{x-10}>0\), we first determined intervals using critical points, then tested them.
For each interval:

• \((-\infty, -10)\) yields a positive result when tested, so it's included in the solution set.
• \((-10, 10)\) yields a negative result, thus excluded from the solution set.
• \((10, \infty)\) also yields a positive result, included in the solution set.

Thus, the solution set becomes the union of \((-\infty, -10) \cup (10, \infty)\), meaning that for all x-values in these intervals, the inequality holds true. Communicating that solution set in mathematics includes both identifying it through critical points and expressing it accurately using interval notation.