Answer :
Let's analyze each of the given inequalities one by one to determine which ones are true for all real numbers:
1. Inequality: \( a - b < a + b \)
Subtract \( a \) from both sides:
[tex]\[ a - b - a < a + b - a \][/tex]
Simplifies to:
[tex]\[ -b < b \][/tex]
This statement is always true because the negative of any number is always less than the number itself for all real numbers \( b \).
Therefore, this inequality is true for all real numbers.
2. Inequality: \( a c \geq b c \)
This inequality depends on the sign of \( c \):
- If \( c > 0 \), multiplying both sides of an inequality \( a \geq b \) by \( c \) retains the inequality direction: \( a c \geq b c \).
- If \( c < 0 \), multiplying both sides by \( c \) reverses the inequality direction: \( a c \leq b c \).
Because the direction of the inequality depends on the sign of \( c \), this inequality is not always true for all real numbers.
3. Inequality: If \( a \geq b \), then \( a + c \geq b + c \)
Add \( c \) to both sides of \( a \geq b \):
[tex]\[ a + c \geq b + c \][/tex]
This statement is always true for any real number \( c \) when \( a \geq b \).
Therefore, this inequality is true for all real numbers.
4. Inequality: If \( c > d \), then \( a - c < a - d \)
Subtract \( a \) from both sides of \( a - c < a - d \):
[tex]\[ a - c - a < a - d - a \][/tex]
Simplifies to:
[tex]\[ -c < -d \][/tex]
Multiply both sides by \(-1\) (which reverses the inequality):
[tex]\[ c > d \][/tex]
This statement matches our initial condition; therefore, this inequality is true for all real numbers when \( c > d \).
However, the statement checks for a specific condition \( c > d \), not all possible real numbers.
5. Inequality: If \( a < b \), then \( a < b + c \)
Subtract \( b \) from both sides:
[tex]\[ a - b < b + c - b \][/tex]
Simplifies to:
[tex]\[ a - b < c \][/tex]
While this could be true depending on the value of \( c \), it is not guaranteed for all real numbers because \( c \) could be negative or positive which affects the validity.
Given our analysis, the detailed conclusion is that:
1. \( a - b < a + b \)
3. If \( a \geq b \), then \( a + c \geq b + c \)
These are the inequalities that are true for all real numbers.
1. Inequality: \( a - b < a + b \)
Subtract \( a \) from both sides:
[tex]\[ a - b - a < a + b - a \][/tex]
Simplifies to:
[tex]\[ -b < b \][/tex]
This statement is always true because the negative of any number is always less than the number itself for all real numbers \( b \).
Therefore, this inequality is true for all real numbers.
2. Inequality: \( a c \geq b c \)
This inequality depends on the sign of \( c \):
- If \( c > 0 \), multiplying both sides of an inequality \( a \geq b \) by \( c \) retains the inequality direction: \( a c \geq b c \).
- If \( c < 0 \), multiplying both sides by \( c \) reverses the inequality direction: \( a c \leq b c \).
Because the direction of the inequality depends on the sign of \( c \), this inequality is not always true for all real numbers.
3. Inequality: If \( a \geq b \), then \( a + c \geq b + c \)
Add \( c \) to both sides of \( a \geq b \):
[tex]\[ a + c \geq b + c \][/tex]
This statement is always true for any real number \( c \) when \( a \geq b \).
Therefore, this inequality is true for all real numbers.
4. Inequality: If \( c > d \), then \( a - c < a - d \)
Subtract \( a \) from both sides of \( a - c < a - d \):
[tex]\[ a - c - a < a - d - a \][/tex]
Simplifies to:
[tex]\[ -c < -d \][/tex]
Multiply both sides by \(-1\) (which reverses the inequality):
[tex]\[ c > d \][/tex]
This statement matches our initial condition; therefore, this inequality is true for all real numbers when \( c > d \).
However, the statement checks for a specific condition \( c > d \), not all possible real numbers.
5. Inequality: If \( a < b \), then \( a < b + c \)
Subtract \( b \) from both sides:
[tex]\[ a - b < b + c - b \][/tex]
Simplifies to:
[tex]\[ a - b < c \][/tex]
While this could be true depending on the value of \( c \), it is not guaranteed for all real numbers because \( c \) could be negative or positive which affects the validity.
Given our analysis, the detailed conclusion is that:
1. \( a - b < a + b \)
3. If \( a \geq b \), then \( a + c \geq b + c \)
These are the inequalities that are true for all real numbers.
