Which inequality is true for all real numbers? Check all that apply.
A. a−b≥c≥b B. ac≥bc C. If a≤b, then a+c≤b+c D. If c>d, then a−c≤b−c E. If a≥b, then a−c≥b−c Note: The exact inequalities in the
The Correct Answer and Explanation is :
To determine which inequalities are true for all real numbers, let’s analyze each option one by one.
A. ( a – b \geq c \geq b )
This statement can be broken down into two parts: ( a – b \geq c ) and ( c \geq b ).
- For ( a – b \geq c ) to hold, it is not universally true because if we choose values such as ( a = 1, b = 2, c = 3 ), then ( 1 – 2 \geq 3 ) is false.
- Similarly, for ( c \geq b ), this does not hold for all real numbers. Thus, this inequality is not true for all real numbers.
B. ( ac \geq bc )
For this inequality to hold for all ( a, b, c ), we can consider the case where ( c = 0 ). In this case, both sides equal zero ((0 \geq 0)). However, if ( c ) is positive and ( a < b ), the inequality becomes ( ac < bc ), which is not valid. If ( c ) is negative and ( a < b ), the inequality becomes ( ac > bc ). Thus, this inequality is not true for all real numbers.
C. If ( a \leq b ), then ( a + c \leq b + c )
This statement is true. If we add the same number ( c ) to both sides of the inequality ( a \leq b ), the direction of the inequality remains unchanged. Therefore, this inequality is true for all real numbers.
D. If ( c > d ), then ( a – c \leq b – c )
Rearranging gives ( a \leq b ). This is not guaranteed to hold for all ( a ) and ( b ). For example, if ( a = 5, b = 3, c = 4, d = 2 ), the inequality ( 5 – 4 \leq 3 – 4 ) translates to ( 1 \leq -1 ), which is false. Thus, this inequality is not true for all real numbers.
E. If ( a \geq b ), then ( a – c \geq b – c )
Similar to option C, if we subtract ( c ) from both sides of the inequality ( a \geq b ), the inequality remains valid. Hence, this inequality is true for all real numbers.
Conclusion
The inequalities that hold true for all real numbers are C and E. These properties are rooted in the basic properties of inequalities, ensuring that adding or subtracting the same value does not affect the relationship between the numbers involved.