Problem 88 - Algebra and Trigonometry

Question

Solve each inequality in Exercises \(86-91\) using a graphing utility. $$ x^{3}+x^{2}-4 x-4>0 $$

Step-by-Step Solution

Answer

The inequality \(x^{3}+x^{2}-4x-4>0\) is solved by identifying the intervals on the cubic function graph where the Y values are greater than 0. The solution is then written in interval notation.

Step 1: Graph the Function

Using a graphing calculator or utility, graph the cubic function \(x^{3}+x^{2}-4x-4\). Identify the x-values at which the function cuts across or touches the x-axis.

Step 2: Analyze the graph and Intervals

Analyze the graph to identify the intervals where the function value \(y\) is greater than 0. This includes the x values to the left and right of where the graph cuts across the x-axis. These intervals are the solutions of the inequality \(x^{3}+x^{2}-4x-4>0\). Do not include the points where the function equals zero.

Step 3: Write the Solution in Interval Notation

After identifying the intervals that satisfy the inequality, write the solution in interval notation. To represent an interval extending indefinitely to the left or right, use negative or positive infinity and round brackets. For an interval between two known x-values, use round brackets for exclusive inequality (>). For inclusive inequality (≥), use square brackets for the endpoints.