Question

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

Short answer

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

Step-by-step solution

- 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.

- 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: \(|a + b| \leq |a| + |b|\).

- 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 \(|a + b|\).

- 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.

Key concepts

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 \(|a + b| \leq\ |a| + |b|\).