Problem 42 - Intermediate Algebra

Question

Solve each inequality. $$ \frac{x+2}{x-4}<0 $$

Step-by-Step Solution

Answer

The solution is \(-2 < x < 4\).

Step 1: Identify the Critical Points

First, identify the critical points of the inequality, which occur when the numerator and the denominator are zero. For the numerator, \(x + 2 = 0\), which gives \(x = -2\). For the denominator, \(x - 4 = 0\), giving \(x = 4\). Thus, the critical points are \(x = -2\) and \(x = 4\).

Step 2: Divide the Number Line

Use the critical points \(x = -2\) and \(x = 4\) to divide the number line into intervals: \((-\infty, -2)\), \((-2, 4)\), and \((4, \infty)\). These intervals will be tested to determine where the inequality \(\frac{x+2}{x-4}<0\) holds true.

Step 3: Test Each Interval

For each interval, choose a test point and substitute it into \(\frac{x+2}{x-4}\) to check the sign:- For \((-\infty, -2)\), choose \(x = -3\): \(\frac{-3+2}{-3-4} = \frac{-1}{-7} > 0\).- For \((-2, 4)\), choose \(x = 0\): \(\frac{0+2}{0-4} = \frac{2}{-4} < 0\).- For \((4, \infty)\), choose \(x = 5\): \(\frac{5+2}{5-4} = \frac{7}{1} > 0\).The inequality holds in the interval \((-2, 4)\).

Step 4: Include or Exclude End Points

Determine if the critical points \(x = -2\) and \(x = 4\) are included in the solution:- At \(x = -2\), the numerator is zero, making the expression \(\frac{0}{x-4} = 0\), not less than \(0\).- At \(x = 4\), the denominator is zero, making the expression undefined.Thus, both endpoints are not included in the solution set.

Step 5: Write the Solution

Since \(\frac{x+2}{x-4}<0\) holds true for \(x\) in the interval \((-2, 4)\) without including the endpoints, the solution to the inequality is \(-2 < x < 4\).

Key Concepts

Understanding Critical Points

In the context of solving inequalities, critical points are those specific values of the variable where the expression either becomes zero or undefined. These points are crucial as they act as boundaries dividing the entire set of possible values (known as the domain) into distinct intervals.

For our inequality involving fractions, we determine critical points by examining the parts of the expression: the numerator and the denominator. For example, for the inequality \(\frac{x+2}{x-4}<0\), we find the critical points by setting the numerator \(x+2\) to zero, which gives us \(x = -2\), and the denominator \(x-4\) to zero, which gives us \(x = 4\).

These critical points \(x = -2\) and \(x = 4\) provide the values where the expression changes its behavior or becomes undefined, and form the basis for dividing the number line into intervals, allowing us to test and find valid solutions.

Visualizing with the Number Line

A number line is a simple and effective tool used to visually represent and analyze inequalities. It allows for a clearer understanding of which parts of the real number line satisfy a given inequality.

Upon identifying critical points, we can use these values to section the number line into separate intervals. In our example, the critical points \(-2\) and \(4\) divide the number line into three main intervals:

• \((-\infty, -2)\)
• \((-2, 4)\)
• \((4, \infty)\)

Each interval is then tested to determine which ones satisfy the inequality \(\frac{x+2}{x-4}<0\). The number line makes it easy to observe where each section starts and ends, and to understand how the inequality behaves as \(x\) moves from one interval to the next.

Identifying Solution Intervals

Solution intervals refer to the ranges on the number line where a given inequality holds true when tested. Determining these intervals often involves selecting test points within each interval and evaluating them in the inequality to see if they make the inequality true.

For the intervals we obtain from \(-2\) and \(4\):

• In \((-\infty, -2)\), a test point like \(x = -3\) results in a positive value, not satisfying \(\frac{x+2}{x-4}<0\).
• In \((-2, 4)\), a test point like \(x = 0\) gives a negative value, satisfying the inequality.
• In \((4, \infty)\), a test point like \(x = 5\) results in a positive value, again not satisfying the inequality.

The only interval where the inequality holds is from \(-2\) to \(4\). However, it is important to remember that endpoints are excluded when they make the expression zero or undefined, hence the solution excludes \(-2\) and \(4\), giving us the open interval \(-2 < x < 4\). This process helps in efficiently identifying the correct solution interval where the inequality is true.