Problem 43 - Beginning and Intermediate Algebra

Question

Solve each rational inequality. Graph the solution set and write the solution in interval notation. $$\frac{5}{z+3} \leq 0$$

Step-by-Step Solution

Answer

The solution to the rational inequality \(\frac{5}{z+3} \leq 0\) is \((-\infty, -3]\). The critical point is -3, and after testing the intervals, the graph of the solution set includes a filled circle at -3 and a line extending to the left towards -∞.

Step 1: Find the critical points

The given inequality is: \[\frac{5}{z+3} \leq 0\] We see that the critical points are the values of z where the rational expression is equal to 0 or undefined. But since the numerator is a constant value 5, it does not equal to 0. The expression is undefined when the denominator is equal to 0: \[z+3=0\] \[z=-3\] Therefore, the critical point is -3.

Step 2: Test the intervals

We now have 2 intervals to test: \((-\infty, -3)\) and \((-3, \infty)\). To test an interval, we'll choose any value inside that interval and plug it into the given inequality. If the inequality is true for that value, then the interval is included in the solution. 1. Test the interval \((-\infty, -3)\): Choose \(z = -4\) \[\frac{5}{(-4)+3} \leq 0 \Rightarrow \frac{5}{-1} \leq 0 \] This inequality is true, so the interval \((-\infty, -3)\) is included in our solution. 2. Test the interval \((-3, \infty)\): Choose \(z = 0\) \[\frac{5}{0 + 3} \leq 0 \Rightarrow \frac{5}{3} \leq 0\] This inequality is false, so the interval \((-3, \infty)\) is not included in our solution.

Step 3: Write the answer in interval notation and graph

Since the given inequality is less than or equal to zero, we include the endpoint -3 in our solution. The interval notation for the solution is: \((-\infty, -3] \) To graph the solution set on a number line: 1. Draw a circle at -3. 2. Since -3 is included in our solution, fill in the circle. 3. Draw a line to the left of the circle towards -∞ to represent the interval \((-\infty, -3]\). Thus, the solution to the rational inequality is \((-\infty, -3].\)

Key Concepts

Critical Points

Critical points are essential when solving rational inequalities as they help identify the segments of the number line that need to be tested. In this context, critical points refer to values of the variable where the expression might change in nature, such as where it could be zero or where it is undefined. For rational expressions, undefined points occur when the denominator is zero.

In the exercise provided, the expression \( \frac{5}{z+3} \) has a critical point at \( z = -3 \) because this value makes the denominator zero, consequently making the expression undefined. To find critical points in rational inequalities:

• Set the numerator equal to zero for rational expressions that equate to zero.
• Set the denominator equal to zero to find where the expression is undefined.

These critical points divide the number line into intervals that need to be tested to understand where the inequality holds true.

Interval Notation

Interval notation is used to describe the set of solutions to inequalities in a concise manner. It gives information about the start and end of intervals and whether these endpoints are included in the solution. A square bracket \( [ \) means the endpoint is included, while a parenthesis \( ( \) indicates it is not included.

For the inequality \( \frac{5}{z+3} \leq 0 \), we found that the solution interval was \((-finity, -3] \). This interval communicates that all numbers less than -3 satisfy the inequality, and -3 itself is included in the solution set.

To write intervals effectively in interval notation:

• Use "\( -\infty \)" or "\( \infty \)" to indicate intervals that go on indefinitely in the negative or positive direction, respectively, always paired with parentheses.
• Use brackets for endpoints if the inequality includes equality (like \( \leq \) or \( \geq \)).

Interval notation is efficient and widely used in mathematics for describing solution sets.

Number Line Graphing

Graphing solution sets on a number line is a visual way to understand where an inequality holds true. Here’s a simple way to graph inequalities using a critical point and intervals.

In this problem, the critical point is \( z = -3 \). This value becomes a significant marker on the number line. Since the solution includes all values less than -3 and -3 itself, the number line graph will:

• Have a solid dot at -3 indicating that -3 is included in the solution.
• Have a line extending leftwards from -3 towards \(-\infty\), showing all those values are part of the solution set.

Using a number line for graphing assists in visualizing the relationship between both variables and fulfills part of representing the inequality comprehensively. By ensuring the inclusion of all necessary elements like closed or open dots, the graph effectively reflects the interval notation \((-fty, -3] \). This method is particularly useful when solving or verifying solutions to rational inequalities.