Basic mathematics

Inequalities. By wavy curve method

When we talk about inequalities in basic mathematics, particularly using the wavy curve method, we're diving into a powerful technique for solving polynomial inequalities. This method helps us visualize the solutions on a number line, making it easier to understand where the inequality holds true. Let's break this down step by step.

Understanding Inequalities

An inequality expresses a relationship between two expressions that are not necessarily equal. For example, we might have an inequality like \( f(x) > 0 \), where \( f(x) \) is a polynomial. The goal is to find the values of \( x \) for which this inequality is satisfied.

What is the Wavy Curve Method?

The wavy curve method is a graphical approach to solving polynomial inequalities. It involves a few key steps:

• Identify the roots: First, we need to find the roots of the polynomial equation \( f(x) = 0 \). These roots will be the points where the polynomial changes its sign.
• Plot the roots on a number line: Once we have the roots, we can mark them on a number line. This helps us visualize the intervals where the polynomial may be positive or negative.
• Determine the sign of the polynomial in each interval: By testing points in each interval created by the roots, we can determine whether the polynomial is positive or negative in those intervals.
• Draw the wavy curve: The wavy curve represents the sign of the polynomial across the number line. It "waves" up and down as it crosses the x-axis at the roots.

Step-by-Step Example

Let’s consider the inequality \( x^3 - 6x^2 + 9x < 0 \). Here’s how we can apply the wavy curve method:

Step 1: Find the Roots

We start by solving the equation \( x^3 - 6x^2 + 9x = 0 \). Factoring gives us:

\( x(x^2 - 6x + 9) = 0 \)

Further factoring yields:

\( x(x - 3)^2 = 0 \)

This gives us the roots \( x = 0 \) and \( x = 3 \) (with a multiplicity of 2).

Step 2: Plot the Roots

On a number line, we mark the points \( 0 \) and \( 3 \). These points divide the number line into three intervals: \( (-\infty, 0) \), \( (0, 3) \), and \( (3, \infty) \).

Step 3: Test the Intervals

Next, we choose test points from each interval:

• For \( (-\infty, 0) \), let’s test \( x = -1 \): \( f(-1) = (-1)^3 - 6(-1)^2 + 9(-1) = -1 - 6 - 9 = -16 < 0 \) (negative).
• For \( (0, 3) \), let’s test \( x = 1 \): \( f(1) = 1 - 6 + 9 = 4 > 0 \) (positive).
• For \( (3, \infty) \), let’s test \( x = 4 \): \( f(4) = 64 - 96 + 36 = 4 > 0 \) (positive).

Step 4: Draw the Wavy Curve

Now, we can visualize the sign changes:

Starting from the left, the curve is negative in \( (-\infty, 0) \), positive in \( (0, 3) \), and remains positive in \( (3, \infty) \). At the roots, since \( x = 3 \) has an even multiplicity, the curve touches the x-axis but does not cross it.

Conclusion

From our analysis, the inequality \( x^3 - 6x^2 + 9x < 0 \) holds true in the interval \( (-\infty, 0) \). Thus, the solution set is \( x < 0 \). The wavy curve method not only provides a clear visual representation but also simplifies the process of determining where the polynomial is positive or negative. This technique is particularly useful for higher-degree polynomials where sign changes can be more complex to analyze.