Problem 56 - Beginning and Intermediate Algebra

Question

Solve each rational inequality. Graph the solution set and write the solution in interval notation. $$\frac{3 z}{z+4} \geq 2$$

Step-by-Step Solution

Answer

The solution set to the rational inequality \(\frac{3z}{z+4} \geq 2\) is \((-\infty, -4) \cup (8, \infty)\).

Step 1: Rewrite the inequality

Subtract 2 from both sides to have 0 on the right side of the inequality: \[\frac{3z}{z+4} - 2 \geq 0\]

Step 2: Identify the critical numbers

Find the critical numbers by setting the left side of the inequality equal to 0: \[\frac{3z}{z+4} - 2 = 0\] Now, find the least common denominator of the terms, which is (z+4). To do this, multiply both the numerator and denominator of 2 by (z+4): \[\frac{3z}{z+4} - \frac{2(z+4)}{z+4} = 0\] Combine the terms in the numerator and solve for z: \[\frac{3z - 2(z+4)}{z+4} = 0\] \[\frac{3z - 2z - 8}{z+4} = 0\] \[\frac{z - 8}{z+4} = 0\] Now we have two critical numbers: z = 8 (solution of the equation) and z = -4 (value makes the denominator 0).

Step 3: Test intervals based on the critical numbers

We have three intervals to test: \(-\infty\) to -4, -4 to 8, and 8 to \(\infty\). We need to test if the inequality holds for some value in each interval: 1. Test z = -5 (Interval: \(-\infty < z < -4\)) \[\frac{3(-5)}{-5+4} - 2 = \frac{-15}{-1} - 2 = 15 - 2 = 13 > 0\] The inequality holds in this interval. 2. Test z = 0 (Interval: \(-4 < z < 8\)) \[\frac{3(0)}{0+4} - 2 = \frac{0}{4} - 2 = -2 < 0\] The inequality does not hold in this interval. 3. Test z = 9 (Interval: \(8 < z < \infty\)) \[\frac{3(9)}{9+4} - 2 = \frac{27}{13} - 2 = \frac{1}{13} > 0\] The inequality holds in this interval.

Step 4: Graph the solution set

The solution set includes the intervals: $$(-\infty, -4) \cup (8, \infty)$$ The graph will show an open circle at z = -4 and z = 8, and a solid line extending from -4 to \(-\infty\) (left side) and from 8 to \(\infty\) (right side).

Step 5: Write the solution in interval notation

The solution can be written in interval notation as: $$(-\infty, -4) \cup (8, \infty)$$

Key Concepts

Interval Notation

Interval notation is a mathematical shorthand to express the set of all numbers between given endpoints. It's particularly useful when dealing with solution sets of inequalities. We use different types of brackets to specify whether an endpoint is included or excluded from a set:

• Round brackets, like \((a, b)\), indicate that the endpoint values \(a\) and \(b\) are not included in the interval. This is called an "open interval".
• Square brackets, such as \([a, b]\), indicate "closed intervals," where the endpoints \(a\) and \(b\) are included in the set.

In the exercise provided, the inequality solution is given as \((-\infty, -4) \cup (8, \infty)\). Here, both \(-4\) and \(8\) are approached but not included, hence the use of round brackets. Infinity (\(\infty\)) is always an open interval because infinity isn't a number we can reach. The union symbol \(\cup\) suggests that the solution consists of two separate parts, allowing us to include numbers from both segments.

Critical Numbers

Critical numbers are values of the variable that either make the function equal to zero or cause it to be undefined (like making a denominator zero). These numbers are essential because they help us identify boundaries in the function's behavior, mainly when dealing with rational inequalities.

• In the given exercise, solving the equation \(\frac{z - 8}{z+4} = 0\) gives the critical number \(z = 8\), which makes the numerator zero.
• The equation's denominator, \(z+4\), equals zero at \(z = -4\), causing the function to become undefined. This is another critical point.

We then use these critical numbers to break the number line into intervals to test within the inequality. These tests help determine where the inequality holds true. By using specific test points within these intervals, as seen in the solution process, we can ascertain which intervals satisfy the inequality condition.

Graphing Solution Sets

Graphing the solution set of an inequality involves presenting the intervals where the inequality holds true on a number line. This visual representation helps to better understand the range of solutions. Here is how we approach it:

• Identify the critical numbers, which divide the number line into distinct intervals. For example, the points \(z = -4\) and \(z = 8\) form the boundaries in our case.
• Test a value from each interval to determine if the inequality is satisfied in that part. These tests establish which parts of the graph represent valid solutions.
• Plot the solutions on the number line by marking intervals. Use open circles at critical points like \(-4\) and \(8\) to indicate they are not part of the solution.
• Draw solid lines or shading to show valid intervals: from \(-\infty\) to \(-4\) and from \(8\) to \(\infty\), indicating where the solution set lies.

This graphical approach reinforces understanding and clearly communicates where the inequality has solutions on the real number line.