Problem 89 - Precalculus Student Solutions Manual 5th

Question

Suppose that \(y=5 x+1\) and we want \(y\) to be within 0.002 unit of \(6 .\) For what values of \(x\) will this be true?

Step-by-Step Solution

Answer

The values of x must be between 0.9996 and 1.0004.

Step 1: Understand the problem

We need to find the values of x for which y is within 0.002 units of 6, meaning we want to solve for x such that \(5x + 1 \) is within \([5.998, 6.002]\).

Step 2: Set the inequality

Set up the inequality for the problem: \[5.998 \leq 5x + 1 \leq 6.002\].

Step 3: Solve the inequality

Solve the left part of the inequality: \5.998 - 1 \leq 5x\. This simplifies to: \[4.998 \leq 5x\]. Then divide by 5 to get: \x \geq 0.9996\. \For the right part: \5x + 1 \leq 6.002\. This simplifies to: \[5x \leq 5.002\]. Then divide by 5 to get: \x \leq 1.0004\. Therefore we have: \0.9996 \leq x \leq 1.0004\.

Key Concepts

linear equations

Linear equations are equations where the highest power of the variable is 1. In this exercise, we have the equation:

y = 5x + 1

This means 'y' changes linearly with 'x'. The '5' is the slope, indicating how steep the line is, and the '1' is the y-intercept, representing where the line crosses the y-axis.

Here’s how to better understand this:

• If 'x' increases by 1, 'y' increases by '5' because the slope equals '5'.
• The equation shifts up by '1' on the y-axis since the y-intercept is '1'.

Grasping this concept helps in solving inequalities further.

inequality solving

Inequality solving involves finding the range of values that satisfy an inequality. In the context of our problem, we need:

5x + 1 to be within [5.998, 6.002]. That’s essentially two inequalities to solve together:

5.998 ≤ 5x + 1 and 5x + 1 ≤ 6.002.

To solve these:

• First, treat each inequality separately.
• For the left one: 5x + 1 ≥ 5.998, subtract 1 from both sides to get 5x ≥ 4.998 and then divide by 5.
• For the right one: 5x + 1 ≤ 6.002, subtract 1 from both sides to get 5x ≤ 5.002 and then divide by 5.

Combining these, you get: 0.9996 ≤ x ≤ 1.0004. Always deal with each part separately.

interval notation

Interval notation is a concise way to describe ranges of values. It's especially useful when expressing solutions for inequalities. Here’s how:

• Brackets [ ] denote that the endpoint is included (like ≤ or ≥).
• Parentheses ( ) denote that the endpoint is not included (like < or >).

Using our exercise: 0.9996 ≤ x ≤ 1.0004 becomes [0.9996, 1.0004].

This format is precise and simple:

• The values inside the brackets are the endpoints.
• This tells us the values that 'x' can take.

Understanding interval notation makes reading and writing solutions much easier!