Answer :
Let's analyze and solve each inequality step by step:
1. Inequality 1: [tex]\(\frac{5}{6} < -\frac{1}{3}\)[/tex]
To compare [tex]\(\frac{5}{6}\)[/tex] and [tex]\(-\frac{1}{3}\)[/tex], let's convert them to decimal form:
- [tex]\(\frac{5}{6} \approx 0.8333\)[/tex]
- [tex]\(-\frac{1}{3} \approx -0.3333\)[/tex]
Clearly, [tex]\(0.8333\)[/tex] is not less than [tex]\(-0.3333\)[/tex]. Therefore, [tex]\(\frac{5}{6} < -\frac{1}{3}\)[/tex] is false.
2. Inequality 2: [tex]\(2 \frac{1}{3} > 2 \frac{1}{6}\)[/tex]
Let's convert the mixed numbers to improper fractions:
- [tex]\(2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}\)[/tex]
- [tex]\(2 \frac{1}{6} = 2 + \frac{1}{6} = \frac{12}{6} + \frac{1}{6} = \frac{13}{6}\)[/tex]
We need to compare [tex]\(\frac{7}{3}\)[/tex] and [tex]\(\frac{13}{6}\)[/tex]:
- [tex]\(\frac{7}{3}\)[/tex] can be written as [tex]\(\frac{14}{6}\)[/tex]
Clearly, [tex]\(\frac{14}{6} > \frac{13}{6}\)[/tex]. Therefore, [tex]\(2 \frac{1}{3} > 2 \frac{1}{6}\)[/tex] is true.
3. Inequality 3: [tex]\(2 < -2 \frac{1}{2}\)[/tex]
Let's convert the mixed number to an improper fraction:
- [tex]\(-2 \frac{1}{2} = -2 - \frac{1}{2} = -2.5\)[/tex]
Clearly, [tex]\(2\)[/tex] is not less than [tex]\(-2.5\)[/tex]. Therefore, [tex]\(2 < -2 \frac{1}{2}\)[/tex] is false.
4. Inequality 4: [tex]\(1 \frac{1}{4} > 1 \frac{1}{3}\)[/tex]
Let's convert the mixed numbers to improper fractions:
- [tex]\(1 \frac{1}{4} = 1 + \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4}\)[/tex]
- [tex]\(1 \frac{1}{3} = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3}\)[/tex]
We need to compare [tex]\(\frac{5}{4}\)[/tex] and [tex]\(\frac{4}{3}\)[/tex]:
- Converting [tex]\(\frac{5}{4}\)[/tex] and [tex]\(\frac{4}{3}\)[/tex] to decimal form:
- [tex]\(\frac{5}{4} = 1.25\)[/tex]
- [tex]\(\frac{4}{3} \approx 1.3333\)[/tex]
Clearly, [tex]\(1.25\)[/tex] is not greater than [tex]\(1.3333\)[/tex]. Therefore, [tex]\(1 \frac{1}{4} > 1 \frac{1}{3}\)[/tex] is false.
So, the only true inequality is:
[tex]\[ 2 \frac{1}{3} > 2 \frac{1}{6} \][/tex]
This corresponds to Inequality 2.
1. Inequality 1: [tex]\(\frac{5}{6} < -\frac{1}{3}\)[/tex]
To compare [tex]\(\frac{5}{6}\)[/tex] and [tex]\(-\frac{1}{3}\)[/tex], let's convert them to decimal form:
- [tex]\(\frac{5}{6} \approx 0.8333\)[/tex]
- [tex]\(-\frac{1}{3} \approx -0.3333\)[/tex]
Clearly, [tex]\(0.8333\)[/tex] is not less than [tex]\(-0.3333\)[/tex]. Therefore, [tex]\(\frac{5}{6} < -\frac{1}{3}\)[/tex] is false.
2. Inequality 2: [tex]\(2 \frac{1}{3} > 2 \frac{1}{6}\)[/tex]
Let's convert the mixed numbers to improper fractions:
- [tex]\(2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}\)[/tex]
- [tex]\(2 \frac{1}{6} = 2 + \frac{1}{6} = \frac{12}{6} + \frac{1}{6} = \frac{13}{6}\)[/tex]
We need to compare [tex]\(\frac{7}{3}\)[/tex] and [tex]\(\frac{13}{6}\)[/tex]:
- [tex]\(\frac{7}{3}\)[/tex] can be written as [tex]\(\frac{14}{6}\)[/tex]
Clearly, [tex]\(\frac{14}{6} > \frac{13}{6}\)[/tex]. Therefore, [tex]\(2 \frac{1}{3} > 2 \frac{1}{6}\)[/tex] is true.
3. Inequality 3: [tex]\(2 < -2 \frac{1}{2}\)[/tex]
Let's convert the mixed number to an improper fraction:
- [tex]\(-2 \frac{1}{2} = -2 - \frac{1}{2} = -2.5\)[/tex]
Clearly, [tex]\(2\)[/tex] is not less than [tex]\(-2.5\)[/tex]. Therefore, [tex]\(2 < -2 \frac{1}{2}\)[/tex] is false.
4. Inequality 4: [tex]\(1 \frac{1}{4} > 1 \frac{1}{3}\)[/tex]
Let's convert the mixed numbers to improper fractions:
- [tex]\(1 \frac{1}{4} = 1 + \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4}\)[/tex]
- [tex]\(1 \frac{1}{3} = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3}\)[/tex]
We need to compare [tex]\(\frac{5}{4}\)[/tex] and [tex]\(\frac{4}{3}\)[/tex]:
- Converting [tex]\(\frac{5}{4}\)[/tex] and [tex]\(\frac{4}{3}\)[/tex] to decimal form:
- [tex]\(\frac{5}{4} = 1.25\)[/tex]
- [tex]\(\frac{4}{3} \approx 1.3333\)[/tex]
Clearly, [tex]\(1.25\)[/tex] is not greater than [tex]\(1.3333\)[/tex]. Therefore, [tex]\(1 \frac{1}{4} > 1 \frac{1}{3}\)[/tex] is false.
So, the only true inequality is:
[tex]\[ 2 \frac{1}{3} > 2 \frac{1}{6} \][/tex]
This corresponds to Inequality 2.
