Problem 61 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Critical Points

To solve the inequality \(\frac{x-5}{x+7} \leq 0\), first identify the critical points where the numerator or denominator is zero. The numerator is zero when \(x-5=0\) which gives \(x=5\). The denominator is zero when \(x+7=0\) which gives \(x=-7\). The domain excludes \(x=-7\) because it makes the denominator zero.

Step 2: Determine Test Intervals

Using the critical points, divide the number line into intervals: \((-\infty, -7)\), \((-7, 5)\), and \((5, \infty)\). These intervals will be tested to determine where the inequality holds true.

Step 3: Evaluate Each Interval

Choose a test point from each interval and substitute it back into \(\frac{x-5}{x+7}\):- For \((-\infty, -7)\), choose \(x=-8\). Substitute: \(\frac{-8-5}{-8+7} = \frac{-13}{-1} = 13 > 0\).- For \((-7, 5)\), choose \(x=0\). Substitute: \(\frac{0-5}{0+7} = \frac{-5}{7} < 0\).- For \((5, \infty)\), choose \(x=6\). Substitute: \(\frac{6-5}{6+7} = \frac{1}{13} > 0\).

Step 4: Analyze the Inequality

The solution to \(\frac{x-5}{x+7} \leq 0\) includes intervals where the expression is negative or zero. Only \((-7, 5)\) is valid. The expression is equal to zero at \(x=5\), so include \(x=5\) in the solution. We exclude \(x=-7\) since the expression is undefined there.

Step 5: Write the Solution

Therefore, the solution to the inequality \(\frac{x-5}{x+7} \leq 0\) is \((-7, 5]\). Include 5 because it makes the expression zero, and exclude -7 because it makes the expression undefined.

Key Concepts

Inequality Solutions

Inequality solutions involve finding the set of values that satisfy an inequality. In this case, the inequality is expressed as a rational expression, \( \frac{x-5}{x+7} \leq 0 \). To solve it, we need to determine where the expression is less than or equal to zero. This involves manipulating the expression, isolating variables, and addressing critical points or exception values where the inequality may not hold.

There are several steps to solve such inequalities:

• Identify where the numerator or denominator equals zero.
• Determine the intervals on the number line based on these critical points.
• Test values from those intervals to find where the inequality holds true.
• Analyze results and ensure those intervals or points make the inequality \( \leq 0 \).

Solutions are written in interval notation, highlighting the precise range of values that satisfy the inequality.

Critical Points

Critical points are essential in solving rational inequalities because they indicate where a rational expression changes its sign. The numerator and denominator can create these points:

• If the numerator equals zero, the expression equals zero.
• If the denominator equals zero, the expression is undefined.

For our specific inequality, \( \frac{x-5}{x+7} \leq 0 \), the critical points are:
- The numerator is zero when \( x-5=0 \), thus \( x=5 \). This shows where the expression could be zero.
- The denominator is zero when \( x+7=0 \), thus \( x=-7 \). This makes the expression undefined, so \( x=-7 \) is excluded from the solution.These points help split the number line into intervals to test different parts for whether they satisfy the inequality.

Number Line Intervals

Number line intervals are segments of the real number line that are established using critical points. They allow us to test different ranges of values against the inequality.

For the inequality \( \frac{x-5}{x+7} \leq 0 \), the critical points at \( x=5 \) and \( x=-7 \) split the number line into intervals:

• \(( -\infty, -7 )\)
• \(( -7, 5 )\)
• \(( 5, \infty )\)

These intervals are then tested using sample points. For instance, assuming a test point within them helps to determine if the inequality is satisfied across the whole interval.

This method ensures we correctly capture all valid solutions including endpoint behavior at zero or undefined points.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. They are central to rational inequalities, as they can change sign based on the values of the input variable, \( x \). Managing these expression requires identifying conditions under which they are zero or undefined.

In the inequality \( \frac{x-5}{x+7} \leq 0 \),

• The expression is equal to zero when the numerator is zero, at \( x=5 \).
• It is undefined when the denominator is zero, at \( x=-7 \).

Rational expressions dictate the structure for setting up intervals on a number line. They are key for analyzing the behavior of the inequality across different values, ensuring the solution represents where the inequality truly holds on its defined domain.