Problem 32 - College Algebra

Question

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x+5}{x+2}<0 $$

Step-by-Step Solution

Answer

The solution set for the inequality \(\frac{x+5}{x+2}<0\) is \((-5, -2) \cup (-2, \infty)\).

Step 1: Identify the Critical Points

Critical points are values that make the fraction undefined or change the sign of the fraction. They are either the solution to the numerator \(x+5 = 0\) or the solution to the denominator \(x+2 = 0\), which are \(x=-5\) and \(x=-2\), respectively.

Step 2: Create a Number Line

On a number line, plot the critical points. These points divide the number line into intervals. Here the intervals are \((-\infty, -5)\), \((-5, -2)\), and \((-2, \infty)\).

Step 3: Test Each Interval

Choose a test point in each interval. If substituting the test point into the inequality makes it true, the interval is part of the solution set. The test points chosen can be \(-6\), \(-3\), and \(0\). Substituting these values one by one into the inequality \(\frac{x+5}{x+2}<0\), we find that the inequality holds true for the intervals \((-5, -2)\) and \((-2, \infty)\).

Step 4: Write the Solution in Interval Notation and Graph it

The solution in interval notation is \((-5, -2) \cup (-2, \infty)\). To graph the solution, plot a number line with open circles at \(-5\) and \(-2\), and shade the intervals to the right of \(-2\) and between \(-5\) and \(-2\).

Key Concepts

Critical Points

Critical points play a vital role in solving rational inequalities. Here, critical points are the values of \( x \) that either make the expression undefined or switch the sign. For the inequality \( \frac{x+5}{x+2}<0 \), the critical points come from setting the numerator \( x+5 = 0 \) and the denominator \( x+2 = 0 \). Solving these gives us the critical points \( x = -5 \) and \( x = -2 \), respectively.

These points help us determine where the inequality might change from positive to negative. Remember, critical points are not necessarily part of the solution. Instead, they guide us in examining each interval they create on the number line. Properly identifying these points is crucial to solving rational inequalities correctly.

Interval Notation

Interval notation is a handy way to describe sets of numbers that represent solutions to inequalities.

In our inequality, once the critical points are identified, they divide the number line into separate intervals. We then test these intervals to determine which part satisfies the inequality. The solution to \( \frac{x+5}{x+2}<0 \) is expressed as \((-5, -2) \cup (-2, \infty)\).

In interval notation:

• Round brackets \(( )\) mean the endpoint is not included.
• Square brackets \([ ]\) mean the endpoint is included.
• "\(\cup\)" signifies the union of sets, combining intervals.

Understanding interval notation ensures that we can quickly read and write solution sets for inequalities.

Number Line

Drawing a number line provides a visual representation that helps us understand the distribution of solutions. To graph the solution of \( \frac{x+5}{x+2}<0 \):

1. Plot the critical points \(-5\) and \(-2\) as open circles since these values make the expression zero or undefined and are not included in the solution.
2. Divide the line into three intervals: \((-\infty, -5)\), \((-5, -2)\), and \((-2, \infty)\).
3. Shade the intervals where the inequality holds true, which are \((-5, -2)\) and \((-2, \infty)\).

Using a number line clarifies which parts of the real number set comply with our inequality, offering a straightforward visual method to verify the solution.

Algebraic Solution

Solving rational inequalities algebraically involves identifying critical points and testing intervals. Here's a step-wise breakdown:

1. Identify critical points by solving \( x+5=0 \) and \( x+2=0 \), resulting in \( x = -5 \) and \( x = -2 \).
2. Create intervals on the number line, divided by these critical points.
3. Test a value from each interval to see if it makes the inequality \( \frac{x+5}{x+2}<0 \) true. For instance, testing \(-6\), \(-3\), and \(0\) confirms which intervals satisfy the inequality.
4. Conclude that the solution is \((-5, -2) \cup (-2, \infty)\).

The algebraic solution requires careful attention to detail, ensuring each step logically follows the previous one, enabling us to find accurate solutions.