91Ó°ÊÓ

Finding zeros of a function is like discovering the places where a graph touches or crosses the x-axis. To find these zeros, you set each factor of the function equal to zero and solve for the variable. In the context of inequalities, zeros play a crucial role in determining the points where the sign of the function might change. For instance, with the function

• \( (2x+3)(3x-1)(x-2) \)

you set each factor to zero:

• \( 2x + 3 = 0 \): Solving gives \( x = -\frac{3}{2} \).
• \( 3x - 1 = 0 \): Solving gives \( x = \frac{1}{3} \).
• \( x - 2 = 0 \): Solving gives \( x = 2 \).

These values are critical because they break the number line into segments where the function's sign can change. Understanding where these changes occur helps you solve inequalities effectively.

Interval notation provides a concise way to describe sets of numbers, particularly solutions to inequalities. It uses parentheses \(( )\) and brackets \([ ]\) to indicate if endpoints are included or excluded. When you solve an inequality, such as \((2x+3)(3x-1)(x-2)<0\),the solution set finds expression in interval notation. Here's how you do it:

• Parentheses \(( )\) indicate that an endpoint is not included in the solution.
• Brackets \([ ]\) mean the endpoint is included.

For this problem, after determining the sign of each interval between \( x = -\frac{3}{2}, \frac{1}{3}, \) and \(2\), you identify which segments satisfy the inequality. The values lie in \((-\frac{3}{2}, \frac{1}{3})\cup(\frac{1}{3}, 2)\).Using parentheses here means these specific points aren't part of the solution. This clear, mathematical language is essential for accurately conveying the solution to inequalities.

Graphing inequalities involves plotting solutions on a number line or coordinate plane. It visually represents the set of values making an inequality true. In this example, the inequality \((2x+3)(3x-1)(x-2)<0\) translates to certain intervals on a number line. First, plot the zeros found earlier at \( x = -\frac{3}{2}, \frac{1}{3}, \) and \(2\).Here’s how you graphically show the solution:

• Draw a number line.
• Mark zeros as distinct points. Since the inequality is strict (<), use open circles to illustrate zeros aren't included.
• Shade the intervals where the inequality holds true: \((-\frac{3}{2}, \frac{1}{3})\) and \((\frac{1}{3}, 2)\).

This approach helps you visualize where values satisfy the given inequality. When learning, connecting the algebraic solution with a graphical representation can deepen understanding. It aligns the abstract concepts of algebra with tangible visual insights.