1.You want to use the interval method to solve a rational inequality. Explain how to find the intervals.
2.Explain how the interval method works.
3.Explain in your own words what a reciprocal function is.
1. To find the intervals using the interval method to solve a rational inequality, follow these steps:
Step 1: Begin by setting the rational expression equal to zero. This means making the numerator of the expression equal to zero.
Step 2: Identify the critical values. These are the values of x for which the numerator or denominator of the rational expression is zero.
Step 3: Place the critical values on a number line, marking them with open circles if the inequality is strict (e.g., < or >) and closed circles if the inequality is non-strict (e.g., ≤ or ≥).
Step 4: Choose test intervals. Pick a value from each interval between the critical values (excluding the critical values themselves) as a test point.
Step 5: Evaluate the rational expression at the chosen test points. Determine the sign of the resulting expression in each interval.
Step 6: Combine the interval signs according to the inequality sign. If the inequality is strict (< or >), keep the intervals where the expression is positive or negative (excluding zeros). If the inequality is non-strict (≤ or ≥), keep the intervals where the expression is non-negative or non-positive (including zeros).
Step 7: Express the solution using interval notation. Write down the intervals obtained in step 6 using interval notation.
2. The interval method is a step-by-step approach used to solve rational inequalities and find the intervals that satisfy the inequality. The method relies on determining the sign of the expression in different intervals to identify where the solution lies. By dividing the number line into intervals and testing points within each interval, we can determine the solution space for the inequality.
3. A reciprocal function is a type of function that maps a given input to its reciprocal. In other words, it is a function where the output is equal to one divided by the input. Reciprocal functions are typically represented in the form f(x) = 1/x. For example, if you have a reciprocal function f(x) = 1/x and you input a value of x = 2, the output will be f(2) = 1/2, which is equal to 0.5. Reciprocal functions have specific characteristics, such as asymptotes at x = 0 and a graph that approaches but never touches the x-axis.