Problem 42 - Beginning and Intermediate Algebra

Question

Solve each rational inequality. Graph the solution set and write the solution in interval notation. $$\frac{3}{v-2}<0$$

Step-by-Step Solution

Answer

The solution to the rational inequality \(\frac{3}{v-2}<0\) lies in the interval \((-\infty,2)\). The graph consists of an open circle at the critical point \(v=2\) and shading the interval to the left of 2 towards negative infinity.

Step 1: Determine the critical points

Identify where the inequality is equal to zero and where the denominator equals zero. These critical points divide the number line into intervals: 1. The inequality \(\frac{3}{v-2}=0\) has no solution since a fraction can only be equal to zero if the numerator is zero. 2. The denominator \(v-2=0\) has a solution at \(v=2\).

Step 2: Test each interval

The critical point \(v=2\) divides the number line into two intervals: \((-\infty, 2)\) and \((2, +\infty)\). Pick a test point from each interval and evaluate the inequality to determine if the solutions lie in that interval. - Test point \(v=1\) for the interval \((-\infty, 2)\): \(\frac{3}{1-2}=\frac{3}{-1}=-3<0\). The inequality is true, so this interval is part of our solution. - Test point \(v=3\) for the interval \((2, +\infty)\): \(\frac{3}{3-2}=\frac{3}{1}=3\), but 3 is not less than 0. So this interval is not part of our solution.

Step 3: Write the solution in interval notation

Since the solution lies in the interval \((-\infty, 2)\), we will write the solution in interval notation as: \((-\infty,2)\).

Step 4: Graph the solution

To graph the solution, we represent the solution interval on the number line: 1. Create an open circle on the critical point \(v=2\), indicating 2 is not part of the solution. 2. Shade the interval from 2 to negative infinity. This indicates that all values less than 2 make the inequality true. Now we have solved the rational inequality, graphed the solution set, and wrote the solution in interval notation.

Key Concepts

Understanding Interval Notation

Interval notation is a shorthand way to express a range of values on the number line. Instead of writing out all the numbers between two points, we use parentheses or brackets to describe the interval.

- Use **parentheses** - Example: \((a, b)\) means all numbers between \(a\) and \(b\), but not including \(a\) and \(b\).- Use **brackets** - Example: \([a, b]\) means all numbers between \(a\) and \(b\), including \(a\) and \(b\) themselves.

In the context of our exercise, the solution is written as \((-\infty, 2)\), indicating that all values less than 2 satisfy the inequality. Note that 2 is not part of the solution, which is why we use a parenthesis.

Graphing Inequalities

Graphing inequalities involves illustrating the range of solutions on a number line. This visual representation helps to clearly show which values satisfy the inequality.

Steps for Graphing:

- **Identify Critical Points:** Look for where the expression equals zero or becomes undefined. In this case, \(v=2\) is a critical point because the denominator becomes zero there.- **Open or Closed Circles:** - Use an **open circle** to indicate that a value is not included in the solution set (like at \(v=2\)). - Use a **closed circle** to show that a value is included in the solution.- **Shade the Interval:** Highlight the portion of the number line that represents the solution. Here, values less than 2, or \((-\infty, 2)\), are shaded, showing the numbers that satisfy the inequality.

Critical Points and Their Importance

Critical points are key values where the behavior of an inequality changes. Finding these points is often your starting step in solving rational inequalities.

In rational inequalities like \(\frac{3}{v-2}<0\), critical points occur where:- The **numerator equals zero**, though in our example it doesn't apply since 3 is never zero.- The **denominator equals zero**, making the expression undefined. Here, \(v-2=0\) gives \(v=2\).

Why Are Critical Points Important?

- They divide the number line into sections or intervals.- Help you decide which intervals make the inequality true by testing sample numbers from each interval.- Identify sections where the function may change from positive to negative or vice versa.

Once you know the critical points, you can apply test points to determine which intervals satisfy the inequality. This helps you structure solutions both graphically and in interval notation.