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Chapter 1: Problem 87

Prove the triangle inequality \(|a+b| \leq|a|+|b| .\)

Short Answer

Expert verified
\(|a+b| \leq |a|+|b|\) is true because absolute values represent non-negative distances, satisfying the triangle inequality.

Step by step solution

01

- Define the Absolute Values

Consider two real numbers, denoted as \(a\) and \(b\). Absolute values are defined as \(|a|\) which is the distance of \(a\) from zero on the number line, and similarly, \(|b|\) is the distance of \(b\) from zero.
02

- Apply the Triangle Inequality for Real Numbers

For any real numbers \(x\) and \(y\), the triangle inequality states: \(|x + y| \,\leq\, |x| + |y|\). Here, let \(x = a\) and \(y = b\). Therefore: \(|a + b| \leq |a| + |b|\).
03

- Understanding the Inequality

When \(a\) and \(b\) have the same sign, the inequality becomes an equality: \(|a + b| = |a| + |b|\). When \(a\) and \(b\) have opposite signs, it is evident that: \(|a + b| < |a| + |b|\). Because absolute values represent distances and distances are always non-negative, the sum of the absolute values \(|a| + |b|\) will always be greater than or equal to the absolute value of their sum \(|a + b|\).
04

- Conclusion

By applying the definition of absolute values and the triangle inequality property, it is concluded that for any two real numbers \(a\) and \(b\), the inequality \(|a + b| \leq |a| + |b|\) always holds true.

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Key Concepts

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

Understanding Absolute Values
Absolute values turn any number into a non-negative value, representing the distance from zero on a number line. For a number \(a\), the absolute value is denoted as \(|a|\). Key properties to remember about absolute values include:
  • The absolute value of a positive number remains the same: \(|a| = a\) if \(a\) is positive.
  • The absolute value of a negative number is the positive version of that number: \(|a| = -a\) if \(a\) is negative.
  • The absolute value of zero is zero: \(|0| = 0\).
Real Numbers
Real numbers (\textbb{R}) include all the numbers on the number line, encompassing both rational numbers (like fractions and integers) and irrational numbers (like \(\frac{\root 2 of 2}\) or \(\frac{\root 2 of 3}\)). These numbers can be positive, negative, or zero. Here are some important features of real numbers:
  • They are used in many areas of mathematics, from basic arithmetic to advanced calculus.
  • Real numbers can be represented as points on an infinitely long line, where each point corresponds to a unique real number.
  • Operations like addition, subtraction, multiplication, and division (except by zero) can be performed on real numbers.
Basics of Inequalities
Inequalities express the relationship between two values, indicating that one is larger or smaller than the other. Important points include:
  • Inequalities use symbols like \(>\) (greater than), \(<\) (less than), \(\geq\) (greater than or equal to), and \(\leq\) (less than or equal to).
  • An inequality statement can be true or false. For example, \(3 < 5\) is true, whereas \(5 < 3\) is false.
  • Inequalities maintain their truth if both sides are added, subtracted, multiplied, or divided by the same positive number. However, multiplying or dividing both sides by a negative number reverses the inequality sign.
Understanding these concepts enhances our ability to solve and manipulate inequalities, including the triangle inequality \(|a + b| \leq\ |a| + |b|\).

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Most popular questions from this chapter

Computing Grades In your Economics 101 class, you have scores of \(68,82,87,\) and 89 on the first four of five tests. To get a grade of B, the average of the first five test scores must be greater than or equal to 80 and less than 90 . (a) Solve an inequality to find the least score you can get on the last test and still earn a \(\mathrm{B}\). (b) What score do you need if the fifth test counts double?

Fat Content Suppose that you order a small McCafeTM chocolate shake ( \(15 \mathrm{~g}\) of fat) and an Artisan Grilled Chicken Sandwich" (7 g of fat) at McDonald's. How many oatmeal raisin cookies can you eat ( \(5 \mathrm{~g}\) of fat) and still keep the total fat content of your meal to no more than \(47 \mathrm{~g} ?\)

A 30.5-ounce can of Hills Bros. \(\begin{array}{llll}\text { coffee } & \text { requires } & 58.9 \pi & \text { square }\end{array}\) inches of aluminum. If its height is 6.4 inches, what is its radius?

Find the real solutions of each equation. Use a calculator to express any solutions rounded to two decimal places. $$ x-4 x^{1 / 2}+2=0 $$

Find the real solutions, if any, of each equation. $$ 4 y^{2}-8 y=3 $$
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