Problem 36 - Algebra and Trigonometry

Question

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ (x-5)(x+4) \geq 0 $$

Step-by-Step Solution

Answer

The solution is \([-4, -4] \cup [5, \infty)\).

Step 1: Identify the Critical Points

Find the values of \( x \) that make the expression \((x-5)(x+4)\) equal to zero. These critical points are found by setting each factor equal to zero: \(x-5=0\) and \(x+4=0\). This gives the critical points \(x=5\) and \(x=-4\).

Step 2: Determine the Test Intervals

The critical points \(x=-4\) and \(x=5\) divide the number line into three intervals: \((-\infty, -4)\), \([-4, 5)\), and \([5, \infty)\).

Step 3: Test Each Interval

Choose a test point from each interval and substitute it into the inequality \((x-5)(x+4) \geq 0\) to check where the inequality holds. For example, test \(x = -5\) for \((-\infty, -4)\), \(x = 0\) for \([-4, 5)\), and \(x = 6\) for \([5, \infty)\).

Step 4: Analyze the Test Results

For \((-\infty, -4)\): \(x = -5\,\rightarrow\, ((-5)-5)((-5)+4) = 25\geq0\), false. \([-4, 5]\): \(x = 0\,\rightarrow\, ((0)-5)((0)+4) = -20\geq0\), false. \([5, \infty)\): \(x = 6\,\rightarrow\, ((6)-5)((6)+4) = 10\geq0\), true.

Step 5: Include Critical Points in the Analysis

Check the inequality at the critical points themselves. At \(x=-4\), \((-4-5)(-4+4)=0\geq0\), true. At \(x=5\), \((5-5)(5+4)=0\geq0\), true.

Step 6: Interpret the Results and Write the Solution

From the testing, the solution set in interval notation where the inequality holds is \([-4, -4] \cup [5, \infty)\).

Step 7: Graph the Solution Set

On the number line, shade the regions corresponding to \([-4, -4]\) and \([5, \infty)\). Use closed circles at \(-4\) and \(5\) to indicate these points are included in the solution.

Key Concepts

Critical Points in Inequalities

Critical points are essential in solving inequalities, especially nonlinear ones. These points are the values that turn the expression into zero. By finding these critical points, we can further analyze the inequality and its solution. In our exercise, the inequality is \[ (x-5)(x+4) \geq 0 \] To find the critical points, we set each factor equal to zero:

• For \( x-5=0 \), the critical point is \( x=5 \).
• For \( x+4=0 \), the critical point is \( x=-4 \).

Here, \( x=5 \) and \( x=-4 \) are crucial because they divide the number line into separate test intervals. Understanding that these points are where the inequality changes its behavior helps to determine where the inequality holds true or false. By examining the values around these critical points, we can determine which regions satisfy the inequality as a whole.

Interval Notation

Interval notation is a convenient way to express a range of values that solve an inequality. It describes the set of numbers between two endpoints. The key is to indicate whether the endpoints are included or excluded. In our exercise, we've identified the solution intervals as follows:

• \([-4, -4]\) includes just the point \(-4\), meaning it's a part of the solution due to intersection at zero.
• \([5, \infty)\) includes the point \(5\) and extends to infinity, showing a continuous range starting from 5.

In interval notation, brackets \([ ]\) mean the endpoint is included, while parentheses \(( )\) mean it is not. This notation is crucial for grasping the solution set's continuity and boundaries. The solution \([-4, -4] \cup [5, \infty)\) captures each interval where the nonlinear inequality holds true. Engaging with interval notation solidifies our understanding of solution ranges, making it easier to discern and communicate solutions clearly.

Graphical Representation of Inequalities

Graphical representation brings inequalities to life on the number line. This visual depiction helps in understanding where an inequality is satisfied. To graph our solution, we follow these steps:1. **Identify key points:** Draw a number line and mark the critical points \(-4\) and \(5\).2. **Shade the solution intervals:**

• Shade the point \(-4\) with a closed circle, indicating it's included in the solution set (-4≤ x ≤ -4).
• Shade the interval from \(5\) to the right towards infinity with a closed circle at \(5\).

3. **Interpreting the graph:** A closed circle suggests the endpoint is part of the solution, as opposed to an open circle. Through this pictorial representation, one can quickly visualize and affirm the solution intervals. Graphing is not just a formality, but a powerful tool to evidence the areas where the inequality holds, ensuring a clear-cut comprehension of the solution's scope.