Problem 46 - Beginning and Intermediate Algebra

Question

Solve each inequality, and graph the solution set. $$ \frac{m+4}{m+5} \geq 2 $$

Step-by-Step Solution

Answer

The solution set is \( (-∞, -6] \cup (-5, ∞) \).

Step 1: Move all terms to one side

Subtract 2 from both sides to set the inequality to zero: \( \frac{m+4}{m+5} - 2 \textbackslashgeq 0 \).

Step 2: Combine the fractions

Combine the fractions under a common denominator: \( \frac{m+4}{m+5} - \frac{2(m+5)}{m+5} \textbackslashgeq 0 \), which simplifies to \( \frac{m+4 - 2(m+5)}{m+5} \textbackslashgeq 0 \).

Step 3: Simplify the numerator

Simplify the numerator: \( m + 4 - 2m - 10 = -m - 6 \). So the inequality becomes \( \frac{-m-6}{m+5} \textbackslashgeq 0 \).

Step 4: Set the critical points

Find the critical points by setting the numerator and denominator to zero: \( -m-6 = 0 \) and \( m+5 = 0 \), giving \( m = -6 \) and \( m = -5 \). These are the critical points.

Step 5: Test the intervals

Determine which intervals satisfy the inequality by testing points on the number line around the critical points \( m = -6 \) and \( m = -5 \). The intervals are: \( (-∞, -6) \), \( (-6, -5) \), and \( (-5, ∞) \).

Step 6: Check intervals

For \( m = -7 \) in \( (-∞, -6) \), \( \frac{-(-7)-6}{-7+5} = \frac{1}{-2} = -\frac{1}{2} \leq 0 \). For \( m = -5.5 \) in \( (-6, -5) \), \( \frac{-(-5.5)-6}{-5.5+5} = \frac{-0.5}{-0.5} = 1 \textbackslashgeq 0 \). For \( m = -4 \) in \( (-5, ∞) \), \( \frac{-(-4)-6}{-4+5} = \frac{-2}{1} = -2 \textbackslashleq 0 \).

Step 7: Solution set

Combining the results, the solution set includes \( (-∞, -6] \cup (-5, ∞) \).

Step 8: Graph the solution set

On a number line, shade the interval from \( -∞ \) to \( -6 \) including \( -6 \), and from \( -5 \) to \( ∞ \), excluding \( -5 \). Place an open circle at \( -5 \) and a closed circle at \( -6 \).

Key Concepts

Critical points

When solving rational inequalities, the concept of 'critical points' is crucial. Critical points are values where the numerator or the denominator of the rational expression is zero. They help us divide the number line into sections where we can test the inequality.
For the inequality \(\frac{m+4}{m+5} \geq 2\), we first rewrite it in a single fraction to identify critical points. Simplifying gives \(\frac{-m-6}{m+5} \geq 0\). The critical points are then found by solving \(-m-6 = 0\) and \(m+5 = 0\).

• Solving \(-m-6 = 0\) gives \(m = -6\).
• Solving \(m+5 = 0\) gives \(m = -5\).

These critical points, \(m = -6\) and \(m = -5\), split the number line into different intervals for testing.

Solution set

The 'solution set' refers to the range of values that satisfy the given inequality. Once we identify the critical points, we need to determine which intervals satisfy the inequality.
For the inequality \(\frac{-m-6}{m+5} \geq 0\), we tested three intervals:

• \(m \in (-∞, -6)\)
• \(m \in (-6, -5)\)
• \(m \in (-5, ∞)\)

By checking these intervals, we found that:

• Values in \((-∞, -6)\) and \((-5, ∞)\) satisfy the inequality.

Therefore, the solution set is \((-∞, -6] \cup (-5, ∞)\).

Interval testing

Interval testing is the method of determining which intervals on the number line satisfy the inequality. After finding critical points, the number line is split into intervals. A test point is picked from each interval to check the inequality.
For our inequality \(\frac{-m-6}{m+5} \geq 0\), intervals are \((-∞, -6)\), \((-6, -5)\), and \((-5, ∞)\). By substituting test points:

• For \(m=-7\) in \((-∞, -6)\): \(\frac{-(-7)-6}{-7+5} = -\frac{1}{2} \leq 0\).
• For \(m=-5.5\) in \((-6, -5)\): \(\frac{-(-5.5)-6}{-5.5+5} = 1 \geq 0\).
• For \(m=-4\) in \((-5, ∞)\): \(\frac{-(-4)-6}{-4+5} = -2 \leq 0\).

This step-by-step checking helps determine which parts of the intervals satisfy the inequality.

Graphing inequalities

Graphing the solution on a number line helps visualize which intervals satisfy the inequality. Once intervals are tested, the solution set can be represented graphically.
To graph \(\frac{m+4}{m+5} \geq 2\), we have the solution set \((-∞, -6] \cup (-5, ∞)\).

• Shade the interval from \( -∞ \) to \(-6 \) including \(-6 \).
• Exclude \(-5 \) by placing an open circle at \(-5 \).
• Shade from \(-5 \) to \(∞ \).

The graph shows a closed circle at \(-6 \) (as it's included in the solution) and an open circle at \(-5 \) (as it's excluded).