Problem 1 - Calculus for Biology and Medicine

Question

Find the two numbers that have distance 4 from \(-1\) by (a) measuring the distances on the real-number line and (b) solving an appropriate equation involving an absolute value.

Step-by-Step Solution

Answer

The two numbers are 3 and -5.

Step 1: Measure distance on the real-number line

To find two numbers that are 4 units away from -1 on the real number line, start by considering two possible directions: one moving to the right (positive direction) and one moving to the left (negative direction).

Step 2: Positive direction

If we move 4 units to the right of -1, we reach the number -1 + 4 = 3.

Step 3: Negative direction

If we move 4 units to the left of -1, we reach the number -1 - 4 = -5.

Step 4: Set up the absolute value equation

The absolute value expression that represents a distance of 4 from -1 is given by |x + 1| = 4, where x represents the unknown number.

Step 5: Solve the absolute value equation (part 1)

An absolute value equation like |x + 1| = 4 has two possible solutions, based on removing the absolute value: either x + 1 = 4 or x + 1 = -4.

Step 6: Solve the linear equation x + 1 = 4

Subtract 1 from both sides to find x. Thus, x = 4 - 1 = 3.

Step 7: Solve the linear equation x + 1 = -4

Subtract 1 from both sides to find x. Thus, x = -4 - 1 = -5.

Key Concepts

Distance on a Real-Number Line

The real-number line is like a vast road stretching infinitely in both directions. Each point on this line represents a real number. To understand distances on this line, imagine standing at point \(-1\) and measuring 4 units in both directions: to the left and to the right.

If you move 4 units to the right from \(-1\), you arrive at \(3\), as \(-1 + 4 = 3\). This movement represents moving in a positive direction.

• Move 4 units right from \(-1\) gets you to \(3\).

Conversely, moving 4 units to the left from \(-1\) results in \(-5\), because \(-1 - 4 = -5\). This movement is towards the negative direction.

• Move 4 units left from \(-1\) gets you to \(-5\).

The beauty of the real-number line is that it allows you to visually understand distances and directions, making math feel more tangible.

Solving Linear Equations

Linear equations are like simple math puzzles. They often involve finding the unknown value, represented by a variable, in our case often noted as \(x\). When working with absolute value equations, you'll encounter situations where you need to solve two separate linear equations.

The absolute value equation \(|x + 1| = 4\) suggests there are two scenarios:

• The first scenario: Solve \(x + 1 = 4\). To do this, subtract 1 from both sides to isolate \(x\): \(x = 4 - 1 = 3\).
• The second scenario: Solve \(x + 1 = -4\). Similarly, subtract 1 from both sides: \(x = -4 - 1 = -5\).

Both of these solutions are valid results of the equation. Linear equations help transform abstract mathematical problems into manageable steps, guiding you methodically to the answer.

Real Number Line Direction

Understanding direction on the real-number line is key to grasping distance concepts. We conventionally move to the right to symbolize positive direction and to the left for negative.

When you solve problems involving distances from a point, always keep these directional movements in mind:

• Positive movement: Increases the number. If you start at \(-1\) and move 4 units to the right, you end up at \(3\).
• Negative movement: Decreases the number. Starting at \(-1\) and going 4 units left leads you to \(-5\).

By integrating these directional insights along with solving equations, you'll navigate real-number line problems with greater ease. It's a harmonious blend of visual and analytical reasoning.