Answered

Welcome to Westonci.ca, your one-stop destination for finding answers to all your questions. Join our expert community now! Explore thousands of questions and answers from a knowledgeable community of experts on our user-friendly platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.

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.
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.