Problem 61 - College Algebra

Question

\(\frac{x-5}{x+7} \leq 0, x \neq-7\)

Step-by-Step Solution

Answer

The solution is \((-7, 5]\).

Step 1: Identify the critical points

To solve the inequality \(\frac{x-5}{x+7} \leq 0\), we first identify the critical points by setting the numerator equal to zero: \(x - 5 = 0\). Solving this gives \(x = 5\). Additionally, the denominator cannot be zero, so \(x + 7 eq 0\), implying \(x eq -7\). These critical points divide the number line into intervals.

Step 2: Determine the intervals

The critical points \(x = 5\) and \(x = -7\) divide the number line into three intervals: \((-\infty, -7)\), \((-7, 5)\), and \((5, \infty)\). We will test a point from each interval to determine where the inequality \(\frac{x-5}{x+7} \leq 0\) holds true.

Step 3: Test intervals around critical points

Choose test points in each of the intervals. For example: 1. Choose \(x = -8\) for \((-\infty, -7)\): \(\frac{-8-5}{-8+7} = \frac{-13}{-1} = 13\), which is positive, not satisfying the inequality.2. Choose \(x = 0\) for \((-7, 5)\): \(\frac{0-5}{0+7} = \frac{-5}{7}\), which is negative, satisfying the inequality.3. Choose \(x = 6\) for \((5, \infty)\): \(\frac{6-5}{6+7} = \frac{1}{13}\), which is positive, not satisfying the inequality.

Step 4: Check inclusion of critical point

Since the inequality \(\leq 0\) is non-strict at zero, check the critical point \(x = 5\): \(\frac{5-5}{5+7} = \frac{0}{12} = 0\), which satisfies the inequality. However, \(x = -7\) is not in the domain of the function.

Step 5: Conclusion on the solution set

The inequality is satisfied on the interval \((-7, 5]\), as this is where \(\frac{x-5}{x+7}\) is less than or equal to zero. Note that \(x = -7\) is excluded because it causes division by zero.

Key Concepts

Critical Points

When you encounter algebraic inequalities involving rational expressions, identifying critical points is an essential first step. Critical points are the values of the variable that make the numerator or denominator equal to zero. These points help you determine the intervals for testing solutions.

In the inequality \(\frac{x-5}{x+7} \leq 0\), we find the critical points by:

• Setting the numerator, \(x-5\), to zero, resulting in \(x = 5\).
• Acknowledging that the expression is undefined when the denominator \(x+7\) equals zero, meaning \(x eq -7\).

These critical points, \(x = 5\) and \(x eq -7\), are vital as they divide the number line into distinct intervals for further analysis.

Interval Notation

After determining the critical points, we utilize them to divide the number line into intervals. This is where the concept of interval notation becomes crucial. Interval notation provides a clear way to write subsets of the real number line.

In our inequality \(\frac{x-5}{x+7} \leq 0\), the critical points \(x = 5\) and \(x = -7\) help form three intervals:

• \((-\infty, -7)\) where \(x\) is less than \(-7\).
• \((-7, 5)\) where \(-7 < x < 5\).
• \((5, \infty)\) where \(x\) is greater than 5.

The notation \((-7, 5]\) indicates all numbers between \(-7\) and \(5\) where \(5\) is included, but \(-7\) is not. This is important because \(x = -7\) causes division by zero, making it undefined.

Rational Expressions

Understanding rational expressions is crucial in handling inequalities like \(\frac{x-5}{x+7}\). A rational expression is a fraction where both the numerator and the denominator are polynomials.

Let's break down the steps:

• The numerator is a simple polynomial: \(x - 5\).
• The denominator is another simple polynomial: \(x + 7\).

The behavior of the rational expression, particularly its sign, changes over the intervals defined by the critical points. It's the transitions through zero that are crucial for finding solutions to inequalities:

• When testing intervals, evaluate the sign of the rational expression. Positive values do not satisfy our inequality of being less than or equal to zero.
• In our solution, since \(\frac{x-5}{x+7}\) is negative or zero between \(-7\) and \(5\), this interval becomes part of the solution.

This comprehension helps in systematically solving and understanding how rational expressions influence inequalities.