Use the graph to solve each equation or inequality. Use interval notation where appropriate. 7x(x - 1)(x - 2) > 0

Verified step by step guidance

1. Identify the roots of the function from the equation \$7x(x - 1)(x - 2) = 0\(. These roots are the values of \)x\( where the function equals zero, which are \)x = 0\(, \)x = 1\(, and \)x = 2$.
2. Use the roots to divide the number line into intervals: \((-\infty, 0)\), \((0, 1)\), \((1, 2)\), and \((2, \infty)\).
3. Determine the sign of the function \$7x(x - 1)(x - 2)$ on each interval by either testing a point from each interval in the function or by analyzing the graph provided.
4. From the graph, identify where the function is greater than zero (i.e., where the graph is above the x-axis). These intervals correspond to the solution of the inequality \$7x(x - 1)(x - 2) > 0$.
5. Express the solution in interval notation by combining the intervals where the function is positive, excluding the points where the function equals zero since the inequality is strict (greater than zero, not greater than or equal to zero).

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality signs. Solving them requires finding intervals where the polynomial is positive or negative, often by analyzing the roots and the sign of the polynomial in each interval.

Roots and Zeros of a Polynomial

The roots or zeros of a polynomial are the values of x where the polynomial equals zero. These points divide the number line into intervals, which are tested to determine where the polynomial is positive or negative, crucial for solving inequalities.

Graph Interpretation for Inequalities

Using the graph of a polynomial function helps visualize where the function is above or below the x-axis. Regions where the graph lies above the x-axis correspond to positive values of the polynomial, aiding in solving inequalities like 7x(x-1)(x-2) > 0.