Question

Tell whether each of the following is true or false. (a) \(-5>-\sqrt{26}\) (b) \(\frac{6}{7}<\frac{34}{39}\) (c) \(-\frac{5}{7}<-\frac{44}{59}\)

Short answer

(a) True, (b) True, (c) False.

Step-by-step solution

Analyze inequality (a)

We have the inequality \(-5 > -\sqrt{26}\). First, evaluate \(-\sqrt{26}\). Since \(\sqrt{26}\) is positive and is approximately 5.099, then \(-\sqrt{26}\) is approximately -5.099. Comparing these two numbers, \(-5 > -5.099\), so the inequality is true.

Analyze inequality (b)

We need to determine if \(\frac{6}{7} < \frac{34}{39}\). Find a common denominator or cross-multiply to compare them. Cross-multiplying: \(6 \times 39 = 234\) and \(7 \times 34 = 238\). Since \(234 < 238\), the inequality \(\frac{6}{7} < \frac{34}{39}\) is true.

Analyze inequality (c)

The inequality to check is \(-\frac{5}{7} < -\frac{44}{59}\). Both fractions have negative numerators, so we need to compare \(\frac{5}{7}\) and \(\frac{44}{59}\). Cross-multiply: \(5 \times 59 = 295\) and \(7 \times 44 = 308\). Since \(295 < 308\), therefore \(-\frac{5}{7} > -\frac{44}{59}\). Thus, \(-\frac{5}{7} < -\frac{44}{59}\) is false.

Key concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. Here are some key points about rational numbers:

• They include both positive and negative numbers.
• The set of rational numbers is denoted by \( \mathbb{Q} \).
• Every integer is a rational number, as it can be expressed as itself divided by 1 (e.g., 5 = \( \frac{5}{1} \)).
• Examples include \( \frac{3}{4}, -\frac{7}{2}, \) and 0.
• Rational numbers can be either terminating or repeating decimals. For instance, \( \frac{1}{4} = 0.25 \) (a terminating decimal), and \( \frac{1}{3} = 0.333... \) (a repeating decimal).

Rational numbers are essential in comparing different quantities, performing arithmetic operations, and solving equations involving fractions. In the exercise, comparing two fractions, such as \( \frac{6}{7} \) and \( \frac{34}{39} \), involves rational numbers.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. Square roots are particularly relevant when working with inequalities involving irrational numbers, such as those in the original exercise. Here are the essentials:

• The square root of a positive number is always non-negative.
• The square root symbol is \( \sqrt{} \).
• For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).
• If we consider negative numbers, \( -\sqrt{9} \) would be -3.
• For numbers that are not perfect squares, like 26, the square root results in an irrational number. For instance, \( \sqrt{26} \approx 5.099 \).

When evaluating inequalities, like in part (a) of the exercise, understanding square roots helps in approximating irrational numbers, thus making comparisons easier.

Cross Multiplication

Cross multiplication is a method used to compare two ratios or fractions by eliminating the division involved in their structure. This technique is handy when dealing with fractions in inequalities or equations. Here's a deeper dive:

• Cross multiplication involves multiplying the numerator of the first fraction by the denominator of the second, and vice versa.
• For example, to compare \( \frac{a}{b} \) and \( \frac{c}{d} \), cross multiply to get \( a \times d \) and \( b \times c \).
• If \( a \times d > b \times c \), then \( \frac{a}{b} > \frac{c}{d} \), and vice versa.
• This method provides a clear and efficient way to compare fractions without having to find a common denominator.

In step 2 of the exercise, cross multiplication was used to determine that \( \frac{6}{7} < \frac{34}{39} \). It shows how comparisons are simplified once the fractions are cross-multiplied.

Negative Numbers

Negative numbers add an extra layer of complexity to both arithmetic operations and the comparison of numbers. They can be represented on a number line below zero and hold unique properties:

• Negative numbers are less than zero.
• The more negative a number is, the smaller its value. For example, -10 is smaller than -5.
• In inequalities, flipping the signs of numbers can reverse the inequality. For instance, if \( a > b \), then \( -a < -b \).

For the exercise analysis, handling negative numbers correctly is crucial, especially when comparing two negative values as seen in parts (a) and (c). For instance, understanding that \(-5 > -\sqrt{26}\) involves recognizing that \(\sqrt{26}\approx5.099\), and thus \(-5 > -5.099\). Moreover, comparing negative fractions (as in part c) demands careful attention to the size of the corresponding positive fractions before flipping the signs.