[Technology Exercise]


a. Graph the functions f(x) = x/2 and g(x) = 1 + (4/x) together to identify the values of x for which


x/2 > 1 + 4/x


b. Confirm your findings in part (a) algebraically.

Verified step by step guidance

1. Step 1: Begin by graphing the functions f(x) = x/2 and g(x) = 1 + (4/x). Use graphing technology or software to plot these functions on the same set of axes. Observe where the graph of f(x) is above the graph of g(x).
2. Step 2: To confirm the graphical findings algebraically, set up the inequality x/2 > 1 + 4/x. This inequality represents the condition where the graph of f(x) is above the graph of g(x).
3. Step 3: Clear the fraction by multiplying both sides of the inequality by x (assuming x ≠ 0) to obtain x^2/2 > x + 4.
4. Step 4: Rearrange the inequality to form a quadratic inequality: x^2/2 - x - 4 > 0. Multiply through by 2 to eliminate the fraction, resulting in x^2 - 2x - 8 > 0.
5. Step 5: Solve the quadratic inequality x^2 - 2x - 8 > 0. Factor the quadratic expression or use the quadratic formula to find the critical points. Determine the intervals where the inequality holds true by testing values within the intervals defined by the critical points.

Key Concepts

Graphing Functions

Graphing functions involves plotting them on a coordinate plane to visually analyze their behavior and intersections. For f(x) = x/2 and g(x) = 1 + (4/x), graphing helps identify where one function is greater than the other by observing their curves and intersection points. This visual representation is crucial for understanding inequalities and relationships between functions.

Inequalities

Inequalities express a relationship where one expression is greater or less than another. In this context, x/2 > 1 + 4/x requires determining the values of x for which the function f(x) is greater than g(x). Solving inequalities often involves algebraic manipulation and understanding the behavior of functions, especially when they involve rational expressions.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve equations or inequalities. To confirm graphically identified solutions algebraically, one must manipulate the inequality x/2 > 1 + 4/x, potentially by finding a common denominator or isolating terms. This process is essential for verifying solutions and understanding the underlying mathematical relationships.