Problem 89 - A Graphical Approach to College Algebra

Question

Solve each problem. Systolic blood pressure is the maximum pressure produced by each heartbeat. Both low blood pressure and high blood pressure are cause for medical concern. Therefore, health care professionals are interested in a patient's "pressure difference from normal," or \(P_{d}\). If 120 is considered a normal systolic pressure, \(P_{d}=|P-120|,\) where \(P\) is the patient's recorded systolic pressure. For example, a patient with a systolic pressure \(P\) of 113 would have a pressure difference from normal of \(P_{d}=|P-120|=|113-120|=|-7|=7\) (a) Calculate the \(P_{d}\) value for a woman whose actual systolic pressure is 116 and whose normal value should be 125 (b) If a patient's \(P_{d}\) value is 17 and the normal pressure for his sex and age should be \(120,\) what are the two possible values for his systolic blood pressure?

Step-by-Step Solution

Answer

(a) 9, (b) 103 and 137

Step 1: Understanding the Formula for Pressure Difference

The formula given for pressure difference is: \[ P_{d} = |P - N| \]where \( P \) is the patient's recorded systolic pressure, and \( N \) is the normal systolic pressure. The absolute value operation means we are interested in the magnitude of the difference, not its sign.

Step 2: Part (a): Calculating Pressure Difference for a Specific Patient

For part (a), we know the patient's actual systolic pressure is 116, and the normal pressure for her is 125. Substitute these values into the formula:\[ P_{d} = |116 - 125| \]Calculate the difference:\[ 116 - 125 = -9 \]Now, take the absolute value to find the pressure difference:\[ P_{d} = |-9| = 9 \]

Step 3: Part (b): Solving for Systolic Pressure Given Pressure Difference

For part (b), the normal value \( N \) is 120. We are given \( P_{d} = 17 \). Use the formula:\[ |P - 120| = 17 \]This equation has two possible solutions by the nature of absolute value, either:1. \( P - 120 = 17 \)2. \( P - 120 = -17 \)

Step 4: Solving the Equations from Part (b)

Solve the two equations:1. For \( P - 120 = 17 \): \[ P = 17 + 120 = 137 \]2. For \( P - 120 = -17 \): \[ P = -17 + 120 = 103 \]Thus, the two possible systolic pressures are 137 and 103.

Key Concepts

Systolic Blood Pressure

Systolic blood pressure refers to the peak pressure in the arteries when the heart beats and pumps blood. It plays a crucial role in overall cardiovascular health. Measuring systolic blood pressure helps assess how much pressure your blood is exerting against artery walls when the heart is contracting.
Both excessively high and low systolic pressures can indicate potential health issues. A typical normal systolic blood pressure is around 120 mmHg, but this can vary based on factors such as age and sex.
Maintaining an appropriate range of systolic blood pressure is vital. Regular checks are advised to ensure it remains in an optimal range, thereby reducing the risk of complications such as heart disease or stroke.

Pressure Difference Formula

The pressure difference in systolic blood pressure can be expressed mathematically using the formula: \[ P_{d} = |P - N| \]where:

• \( P_{d} \) is the pressure difference from normal.
• \( P \) is the recorded systolic pressure of the patient.
• \( N \) is the normal pressure for the individual's age and sex.

This formula utilizes the concept of absolute value, which ensures that the result is a non-negative magnitude, reflecting the true difference devoid of direction (positive or negative).
Understanding this formula allows health professionals to assess deviations from normal blood pressure efficiently, which is crucial for early detection and management of potential health problems.

Magnitude of Difference

In the pressure difference formula, the concept of magnitude is represented by the absolute value, denoted as \(|x|\). This mathematical operation considers the size of the difference regardless of its sign.
For example, if a patient's systolic pressure is less than the normal, the difference might be negative. However, the absolute value turns this negative into a positive, emphasizing the magnitude alone.
Wherever precise evaluation of a deviation is necessary, such as in medical assessments like blood pressure readings, focusing on the magnitude provides a clearer understanding. A differing magnitude informs healthcare providers of how far a patient's pressure is from the normal, enabling them to take suitable action.