Problem 56 - College Algebra

Question

A patient is not allowed to have more than 330 milligrams of cholesterol per day from a diet of eggs and meat. Each egg provides 165 milligrams of cholesterol. Each ounce of meat provides 110 milligrams. a. Write an inequality that describes the patient's dietary restrictions for \(x\) eggs and \(y\) ounces of meat. b. Graph the inequality. Because \(x\) and \(y\) must be positive, limit the graph to quadrant I only. c. Select an ordered pair satisfying the inequality. What are its coordinates and what do they represent in this situation?

Step-by-Step Solution

Answer

a. The inequality that describes the patient's dietary restriction is \(165x + 110y \leq 330\). b. The graph should show a line originating at the point (3,0) and going through the point (0,2), with the area below the line shaded, and restricted to quadrant 1. c. For example, the ordered pair (1,1) represents a dietary choice of 1 egg and 1 ounce of meat for the day.

Step 1: Formulate the Inequality

Given that \(x\) is the number of eggs and \(y\) is the number of ounces of meat, and that each egg provides 165 milligrams of cholesterol and each ounce of meat provides 110 milligrams. The total cholesterol can be expressed as \(165x + 110y\). Since the patient should not exceed 330 milligrams of cholesterol per day, we formulate the inequality as \(165x + 110y \leq 330\).

Step 2: Graph the Inequality

To graph the inequality \(165x + 110y \leq 330\), first put the inequality into slope-intercept form \(y \leq -1.5x + 3\). The slope of the line is -1.5 and the y-intercept is 3. Plot the y-intercept and use the slope to find another point on the line. Draw the line and fill in the area below the line because the sign of the inequality is \(\leq\), indicating the related points are on or below the line. Because both x (eggs) and y (ounces of meat) have to be nonnegative, we only consider quadrant I of the coordinate system.

Step 3: Interpret an Ordered Pair

Let's choose an arbitrary ordered pair that lies in the shaded region, such as (1,1). This implies that the patient can have 1 egg and 1 ounce of meat without exceeding the cholesterol limit. Remember that the x-coordinate represents the number of eggs and the y-coordinate represents the number of ounces of meat.