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 equation from the given polynomial \$7x(x - 1)(x - 2) = 0\(. The roots are the values of \)x\( that make the equation equal to zero. Set each factor equal to zero: \)x = 0\(, \)x - 1 = 0 \Rightarrow x = 1\(, and \)x - 2 = 0 \Rightarrow x = 2$.
2. These roots divide the number line into four intervals: \((-\infty, 0)\), \((0, 1)\), \((1, 2)\), and \((2, \infty)\). We will analyze the sign of the polynomial on each interval to solve inequalities.
3. To solve the equation \$7x(x - 1)(x - 2) = 0\(, note that the solutions are exactly the roots found: \)x = 0\(, \)x = 1\(, and \)x = 2$.
4. To solve inequalities such as \$7x(x - 1)(x - 2) > 0\( or \)7x(x - 1)(x - 2) < 0$, test a sample point from each interval in the polynomial to determine if the polynomial is positive or negative in that interval.
5. Use the graph to confirm the sign of the polynomial on each interval: where the graph is above the \(x\)-axis, the polynomial is positive; where it is below, the polynomial is negative. Then write the solution in interval notation based on the inequality.

Key Concepts

Roots of a Polynomial

The roots of a polynomial are the values of x where the polynomial equals zero. For the equation 7x(x - 1)(x - 2) = 0, the roots are x = 0, x = 1, and x = 2. These points correspond to where the graph intersects the x-axis.

Solving Polynomial Inequalities Using Graphs

To solve inequalities involving polynomials, analyze the graph to determine where the function is above or below the x-axis. For example, to solve 7x(x - 1)(x - 2) > 0, find intervals where the graph is positive (above the x-axis). Use interval notation to express these solution sets.

Interval Notation

Interval notation is a concise way to represent sets of numbers between two endpoints. Parentheses () indicate that endpoints are not included, while brackets [] mean they are included. This notation is used to express solution sets for equations and inequalities clearly.