91Ó°ÊÓ

Polynomial inequalities are expressions that involve a polynomial on one side of an inequality symbol, such as <, >, ≤, or ≥. These inequalities require us to find the set of values for the variable that makes the inequality true. For the given exercise, the polynomial inequality is \( (x+1)(x-5)>0 \).
Working with polynomial inequalities involves several steps:

• Identify the roots of the polynomial by setting the expression equal to zero.
• Determine the intervals into which these roots divide the number line.
• Use test points to check the sign of the polynomial in each interval.

Understanding polynomial inequalities is crucial because it helps us determine the range of values for which the inequality holds true. These techniques are widely used in various fields, including engineering, economics, and science.

Interval testing is a method used to solve polynomial inequalities by analyzing the sign of the polynomial expression within different intervals. The key steps are:
1. **Find the Roots**: Start by solving \( (x+1)(x-5)=0\). In this exercise, we find that \(x = -1\) and \(x = 5\).
2. **Divide the Number Line**: The roots \(x = -1\) and \(x = 5\) divide the number line into three intervals: \(x < -1, -1 < x < 5, x > 5\).
3. **Test Each Interval**: Choose a sample point from each interval and substitute it into the polynomial inequality to see if it holds true:

• For \(x < -1\), choose \(x = -2\); \((x+1)(x-5) > 0 \) is true.
• For \(-1 < x < 5\), choose \(x = 0\); \((x+1)(x-5) > 0 \) is false.
• For \(x > 5\), choose \(x = 6\); \((x+1)(x-5) > 0 \) is true.

Using interval testing helps clarify where the polynomial inequality holds true on the number line, making it easier to determine the solution set.

Graphing the solutions of a polynomial inequality involves visually representing the solution set on a number line. Here’s how to do it step-by-step:
1. **Identify the Critical Points**: These are the roots or zeros of the polynomial. For this exercise, the critical points are \(-1\) and \(5\).
2. **Plot the Critical Points**: On a number line, mark the roots with open circles because the inequality \(>\) does not include the points themselves.
3. **Shade the Solution Areas**: Based on interval testing, shade the regions where the polynomial inequality is true. Here, you’ll shade the regions for \(x < -1\) and \(x > 5\).
4. **Check Your Work**: Verify that the shading corresponds to the true intervals determined from the interval tests.

• For \(x < -1\), the area to the left of \(-1\) is shaded.
• For \(x > 5\), the area to the right of \(5\) is shaded.

Graphing solutions provides a clear visual representation of the intervals where the polynomial inequality holds true, reinforcing the findings from interval testing.