Question
given that 0 < c < 1 what inequalities must be true
Answers
GPT-4o mini
Given the condition \( 0 < c < 1 \), we can derive several useful inequalities involving the variable \( c \):
1. **Multiplicative Inequalities**:
- For any positive number \( x \):
\[
0 < cx < x
\]
This is valid because multiplying a positive number by a fraction less than 1 results in a smaller positive number.
2. **Sum Inequalities**:
- For any positive numbers \( a \) and \( b \):
\[
a < a + c < a + 1
\]
3. **Subtraction Inequalities**:
- For any \( x \) greater than \( c \):
\[
x - 1 < x - c < x
\]
4. **Zero and Negatives**:
- For any negative number \( y \):
\[
cy < 0
\]
5. **Divisional Inequalities**:
- If \( x > 0 \), then:
\[
c < \frac{c}{x} < 1
\]
These inequalities are a direct result of the property that \( c \) is a positive fraction less than 1. They are useful in various mathematical contexts, such as optimization problems, calculus, and algebra.
1. **Multiplicative Inequalities**:
- For any positive number \( x \):
\[
0 < cx < x
\]
This is valid because multiplying a positive number by a fraction less than 1 results in a smaller positive number.
2. **Sum Inequalities**:
- For any positive numbers \( a \) and \( b \):
\[
a < a + c < a + 1
\]
3. **Subtraction Inequalities**:
- For any \( x \) greater than \( c \):
\[
x - 1 < x - c < x
\]
4. **Zero and Negatives**:
- For any negative number \( y \):
\[
cy < 0
\]
5. **Divisional Inequalities**:
- If \( x > 0 \), then:
\[
c < \frac{c}{x} < 1
\]
These inequalities are a direct result of the property that \( c \) is a positive fraction less than 1. They are useful in various mathematical contexts, such as optimization problems, calculus, and algebra.