Problem 29 - Algebra and Trigonometry

Question

\(29-32\) me state whether each inequality is true or false. \(\begin{array}{ll}{\text { (a) }-6<-10} & {\text { (b) } \sqrt{2}>1.41}\end{array}\)

Step-by-Step Solution

Answer

(a) False, (b) True.

Step 1: Understanding Inequalities

In both statements, we need to determine the truth of a comparison between two numbers. An inequality like \(-6 < -10\) is asking if \(-6\) is less than \(-10\), while \(\sqrt{2} > 1.41\) is checking if the square root of 2 is greater than \(1.41\).

Step 2: Evaluating Negative Numbers

Compare the integers \(-6\) and \(-10\) on a number line. Note that \(-6\) is to the right of \(-10\) on the number line, meaning \(-6 > -10\). Thus, the inequality \(-6 < -10\) is false.

Step 3: Calculating Square Roots

Calculate or use known approximations of \(\sqrt{2}\). We know \(\sqrt{2} \approx 1.414\). Compare this value with \(1.41\) and observe that \(1.414 > 1.41\). Hence, the inequality \(\sqrt{2} > 1.41\) is true.

Key Concepts

Comparing Numbers on a Number Line

Understanding how to compare numbers using a number line is crucial for evaluating inequalities. A number line is a straight line with numbers placed at equal intervals along its length. The further right a number is located on the number line, the larger its value. This applies to both positive and negative numbers.

When dealing with inequalities that involve negative numbers, the right-left positioning is incredibly important. For instance, consider comparing

• \(-6\)
• \(-10\)

Based on number line logic, since

• \(-6\) is located to the right of \(-10\)

We conclude that

• \(-6\) is greater than \(-10\)

Thus, in the inequality \(-6 < -10\), the statement is false because the positioning on the number line contradicts the inequality symbol. Understanding this concept allows you to confidently tackle similar problems.

Square Roots Approximations

Approximating square roots can be exceptionally helpful in verifying the truth of an inequality involving irrational numbers. Square roots of non-perfect squares, such as 2, can be irrational numbers, meaning they can't be expressed as a simple fraction and often require approximation.

For example, if you need to assess the statement \(\sqrt{2} > 1.41\), it’s vital to understand that:

• \(\sqrt{2} \approx 1.414\)

This approximation reveals that

• \(1.414 > 1.41\)

Hence,

• the inequality \(\sqrt{2} > 1.41\) is indeed true.

Being familiar with common square root approximations will significantly enhance your ability to evaluate similar expressions swiftly.

Evaluating Negative Numbers

When determining the truth of inequalities with negative numbers, it helps to remember that negative numbers closer to zero are greater. Evaluating inequalities such as \(-6 < -10\) requires careful consideration of their positions relative to zero.

In this case:

• -6 is greater than -10

To visualize this, imagine that both numbers are points on a number line:

• \(-6\) is to the right of \(-10\)

Since numbers further to the right (closer to zero) are of greater value in the realm of negative numbers, the inequality \(-6 < -10\) is false.

By consistently practicing these evaluations, you can develop an intuitive understanding of how negative numbers interact with inequality signs.