Compare the following values using the symbols <, >, or =. a. -2 _______ -|-2| b. 2/3 _______ 3/4 c. -1.4 ______ |-1.3| d. 11/8 _______ −11/8
Understand the Problem
The question asks us to compare pairs of numerical values, including integers, fractions, and decimals (including their absolute values), and determine the correct relationship between them. We will use the symbols '<' for 'less than', '>' for 'greater than', or '=' for 'equal to.' For each pair, we'll evaluate both sides and then compare the values.
Answer
(a) $-5 < 2$ (b) $\frac{2}{3} > \frac{3}{5}$ (c) $|-6| > |-4|$ (d) $0.125 = \frac{1}{8}$ (e) $-1.5 = -\frac{3}{2}$
Answer for screen readers
(a) $-5 < 2$ (b) $\frac{2}{3} > \frac{3}{5}$ (c) $|-6| > |-4|$ (d) $0.125 = \frac{1}{8}$ (e) $-1.5 = -\frac{3}{2}$
Steps to Solve
- Compare $-5$ and $2$
Since negative numbers are always less than positive numbers, $-5$ is less than $2$. Therefore, $-5 < 2$.
- Compare $\frac{2}{3}$ and $\frac{3}{5}$
To compare fractions, we can find a common denominator. The least common denominator for $3$ and $5$ is $15$. Convert both fractions to equivalent fractions with the denominator $15$.
$\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}$
$\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}$
Now compare the numerators: $10 > 9$. Therefore, $\frac{10}{15} > \frac{9}{15}$, which means $\frac{2}{3} > \frac{3}{5}$.
- Compare $|-6|$ and $|-4|$
The absolute value of a number is its distance from $0$.
$|-6| = 6$
$|-4| = 4$
Since $6 > 4$, we have $|-6| > |-4|$.
- Compare $0.125$ and $\frac{1}{8}$
Convert the fraction to a decimal: $\frac{1}{8} = 0.125$
Since $0.125 = 0.125$, we have $0.125 = \frac{1}{8}$.
- Compare $-1.5$ and $-\frac{3}{2}$
Convert the fraction to a decimal: $-\frac{3}{2} = -1.5$
Since $-1.5 = -1.5$, we have $-1.5 = -\frac{3}{2}$.
(a) $-5 < 2$ (b) $\frac{2}{3} > \frac{3}{5}$ (c) $|-6| > |-4|$ (d) $0.125 = \frac{1}{8}$ (e) $-1.5 = -\frac{3}{2}$
More Information
The absolute value of a number is its distance from zero. Converting fractions to decimals or finding a common denominator are useful techniques for comparison.
Tips
A common mistake is to incorrectly compare fractions without converting them to a common denominator or converting them to decimals. Also, students might confuse the rules for comparing negative numbers. For example, $-5$ is less than $2$, not greater. With absolute values, avoid skipping the step of evaluating the absolute value, and then compare.
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