Problem 30 - Algebra and Trigonometry

Question

\(29-32\) me state whether each inequality is true or false. \(\begin{array}{ll}{\text { (a) } \frac{10}{11}<\frac{12}{13}} & {\text { (b) }-\frac{1}{2}<-1}\end{array}\)

Step-by-Step Solution

Answer

(a) True, (b) False.

Step 1: Compare Numerators and Denominators

For the inequality \(\frac{10}{11} < \frac{12}{13}\), compare the numerators (10 and 12) and denominators (11 and 13). Because both numerators and denominators increase, calculate the cross products to compare: \(10 \times 13\) and \(11 \times 12\).

Step 2: Cross Multiplication for (a)

Calculate the cross products: \(10 \times 13 = 130\) and \(11 \times 12 = 132\). Since 130 is less than 132, \(\frac{10}{11} < \frac{12}{13}\) is true.

Step 3: Analyze the Second Inequality Part

For \(-\frac{1}{2} < -1\), remember that on the number line, any negative fraction with a smaller absolute value is greater. Here, \(-\frac{1}{2}\) is greater than \(-1\) because it is less negative.

Step 4: Conclusion for (b)

The statement \(-\frac{1}{2} < -1\) is false because \(-\frac{1}{2}\) is actually greater than \(-1\).

Key Concepts

Fractions

Fractions represent a part of a whole. Each fraction consists of a numerator and a denominator. The numerator is the number on top and shows how many parts there are. The denominator is the number on the bottom and shows the total number of equal parts.
Understanding fractions is vital for comparing the parts of different wholes.
If we want to compare fractions, we need them to relate to the same whole or transform them into a common denomination.

• For example, let's compare \( \frac{10}{11} \) and \( \frac{12}{13} \).
• The numerators are 10 and 12, indicating parts we have.
• The denominators 11 and 13 represent the total pieces we are considering.

The comparison might seem tricky, so we use other methods like cross multiplication to make it easier.

Cross Multiplication

Cross multiplication is a method used to compare two fractions without finding a common denominator. This method is beneficial when given fractions like \( \frac{10}{11} \) and \( \frac{12}{13} \).
To apply this, multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa.
This will give you two products to compare directly.

• For \( \frac{10}{11} < \frac{12}{13} \):
• Multiply across: \( 10 \times 13 = 130 \).
• Then, multiply the other way: \( 11 \times 12 = 132 \).

Since 130 is less than 132, we confirm \( \frac{10}{11} < \frac{12}{13} \). Cross multiplication helps make sense of fractional comparisons quickly and effectively.

Number Line

A number line illustrates numbers in order, making it easier to visualize their relationships, especially when dealing with negative numbers and fractions.
On a number line, numbers to the right are always larger than numbers to the left.
This tool is essential when considering inequalities such as \( -\frac{1}{2} < -1 \).

• Place \( -1 \) and \( -\frac{1}{2} \) on the number line.
• Moving left, \( -1 \) falls before \( -\frac{1}{2} \).

Thus, \( -\frac{1}{2} \) is actually greater than \( -1 \) because it's less negative, showing the importance of understanding placement on a number line. Representing these numbers visually helps clarify their size relative to each other.

Negative Numbers

Negative numbers are numbers less than zero, and they often represent loss or decrease. When working with negative numbers in inequalities, the usual rules seem flipped.
Being 'less' actually means being more negative.

• Think about \( -\frac{1}{2} < -1 \).
• On a number line, we find \( -1 \) is further to the left than \( -\frac{1}{2} \).

Therefore, \( -\frac{1}{2} \) is greater than \( -1 \).
In the real world, losing half of something (e.g., half a dollar) is better than losing one whole (like one dollar).
Understanding negative numbers helps correctly interpret inequalities and real-life scenarios where these numbers come into play.