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Chapter 17: Problem 11

for the inequality \(4<9,\) state the inequality that results when the given operations are performed on both members. Square both.

Short Answer

Expert verified
The resulting inequality is \(16 < 81\).

Step by step solution

01

Understand the Problem

The inequality to begin with is \(4 < 9\). Our task is to square both sides of the inequality.
02

Perform the Operation

Square both sides of the inequality: \(4^2 < 9^2\).
03

Calculate the Squares

Calculate the squares: \(4^2 = 16\) and \(9^2 = 81\).
04

Write the New Inequality

After performing the squaring operation, the new inequality is \(16 < 81\).

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Key Concepts

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

Squaring Numbers: The Foundation
Squaring numbers is an essential part of many mathematical operations. When we talk about squaring a number, we simply mean multiplying a number by itself.
For example, to square the number 4, we calculate:
  • The square of 4 is calculated as: \[4^2 = 4 \times 4 = 16\]
Understanding how to square a number is important when dealing with inequalities because we must apply the operation consistently to both members of the inequality to maintain its truth.
In the context of inequalities, squaring helps us understand how the sizes of numbers relate to each other, even after the operation. Remember, while squaring positive numbers or zero retains the original inequality, squaring negative numbers can reverse it due to the sign change.
Understanding Mathematical Operations in Inequalities
Mathematical operations such as addition, subtraction, multiplication, and division allow us to manipulate inequalities in meaningful ways.
When dealing with inequalities, it is crucial to apply the same operation to both sides to keep the inequality intact. Let鈥檚 discuss some rules:
  • When you add or subtract the same number from both sides of an inequality, its direction does not change.
  • Multiplying or dividing both sides of an inequality by a positive number also keeps the inequality direction unchanged.
  • However, multiplying or dividing both sides by a negative number reverses the inequality direction.
Squaring both sides of an inequality such as \(4 < 9\) means treating each side separately:
  • Square 4 to get 16
  • Square 9 to get 81
The inequality remains true because both sides were squared, keeping the context of comparison consistent.
Step-by-Step Solutions: Making the Process Clear
Solving inequalities, especially when involving operations such as squaring, benefits from a step-by-step approach.
Let鈥檚 revisit the given problem of squaring an inequality:
  • **Step 1**: Identify the initial inequality which is \(4 < 9\). This gives us the starting point.
  • **Step 2**: Perform the required mathematical operation equally on both sides. In this case, square both sides: \(4^2 < 9^2\).
  • **Step 3**: Calculate the squares: \(4^2 = 16\) and \(9^2 = 81\).
  • **Step 4**: State the new inequality based on these calculations: \(16 < 81\).
Following these steps makes sure that each part of the problem is addressed clearly.
It also provides a logical checklist to verify the accuracy of your results, ensuring no steps are missed.

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Most popular questions from this chapter

some applications of inequalities are shown. The electric intensity \(E\) within a charged spherical conductor is zero. The intensity on the surface and outside the sphere equals a constant \(k\) divided by the square of the distance \(r\) from the center of the sphere. State these relations for a sphere of radius \(a\) by using inequalities, and graph \(E\) as a function of \(r\)

Answer the given questions by solving the appropriate inequalities. The weight \(w\) (in \(\mathrm{N}\) ) of an object \(h\) meters above the surface of earth is \(w=r^{2} w_{0} /(r+h)^{2},\) where \(r\) is the radius of earth and \(w_{0}\) is the weight of the object at sea level. Given that \(r=6380 \mathrm{km},\) if an object weighs \(200 \mathrm{N}\) at sea level, for what altitudes is its weight less than \(100 \mathrm{N} ?\)

Use inequalities to solve the given problems. For what values of real numbers \(a\) and \(b\) does the inequality \((x-a)(x-b)<0\) have real solutions?

Use inequalities involving absolute values to solve the given problems. The temperature \(T\) (in \(^{\circ} \mathrm{C}\) ) at which a certain machine can operate properly is \(70 \pm 20 .\) Express the temperature \(T\) for proper operation using an inequality with absolute values.

Set up the necessary inequalities and sketch the graph of the region in which the points satisfy the indicated system of inequalities. A telephone company is installing two types of fiber-optic cable in an area. It is estimated that no more than 300 m of type A cable, and at least 200 m but no more than 400 m of type \(B\) cable, are needed. Graph the possible lengths of cable that are needed.
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