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Chapter 1: Problem 89

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?

Short Answer

Expert verified
The values of x must be between 0.9996 and 1.0004.

Step by step solution

01

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 \([5.998, 6.002]\).
02

Set the inequality

Set up the inequality for the problem: \[5.998 \leq 5x + 1 \leq 6.002\].
03

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\.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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鈥檚 essentially two inequalities to solve together:

5.998 鈮� 5x + 1 and 5x + 1 鈮� 6.002.

To solve these:
  • First, treat each inequality separately.
  • For the left one: 5x + 1 鈮� 5.998, subtract 1 from both sides to get 5x 鈮� 4.998 and then divide by 5.
  • For the right one: 5x + 1 鈮� 6.002, subtract 1 from both sides to get 5x 鈮� 5.002 and then divide by 5.
Combining these, you get: 0.9996 鈮� x 鈮� 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鈥檚 how:

  • Brackets [ ] denote that the endpoint is included (like 鈮� or 鈮�).
  • Parentheses ( ) denote that the endpoint is not included (like < or >).
Using our exercise: 0.9996 鈮� x 鈮� 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!

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Most popular questions from this chapter

Solve each equation or inequality. $$\left|2 x+\frac{1}{3}\right|+1<4$$

Complex numbers are used to describe current I, voltage \(E,\) and impedance \(Z\) (the opposition to current). These three quantities are related by the equation \(E=I Z, \quad\) which is known as Ohm's Law. Thus, if any two of these quantities are known, the third can be found. In each exercise, solve the equation \(E=I Z\) for the remaining value. $$E=57+67 i, Z=9+5 i$$

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Solve each rational inequality. Write each solution set in interval notation. $$\frac{10}{3+2 x} \leq 5$$

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