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Which inequality is true for all real numbers? Check all that apply.

A. [tex]a - b \ \textless \ a + b[/tex]

B. [tex]a c \geq b c[/tex]

C. If [tex]a \geq b[/tex], then [tex]a + c \geq b + c[/tex]

D. If [tex]c \ \textgreater \ d[/tex], then [tex]a - c \ \textless \ a - d[/tex]

E. If [tex]a \ \textless \ b[/tex], then [tex]a \ \textless \ b + c[/tex]

Sagot :

GinnyAnswer
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.
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