ε·²η₯ $|a|=5,|b|=3$οΌδΈ $\mid a-$ $b \mid=b-a$οΌε $a+b=$
Answer (2)
The values of a and b are determined to be a = β 5 and b = 3 , respectively. Therefore, the result of a + b is β 2 .
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The problem states that β£ a β£ = 5 , β£ b β£ = 3 , and β£ a β b β£ = b β a .
The condition β£ a β b β£ = b β a implies a β€ b .
Since β£ a β£ = 5 and β£ b β£ = 3 , the possible values are a = Β± 5 and b = Β± 3 .
Given a β€ b , we determine that a = β 5 and b = 3 , so a + b = β 5 + 3 = β 2 β .
Explanation
Understanding the Problem We are given that β£ a β£ = 5 , β£ b β£ = 3 , and β£ a β b β£ = b β a . We need to find the value of a + b .
Analyzing the Absolute Value Condition The condition β£ a β b β£ = b β a implies that a β b is non-positive, i.e., a β b β€ 0 , which means a β€ b .
Possible Values of a and b Since β£ a β£ = 5 , a can be either 5 or β 5 . Since β£ b β£ = 3 , b can be either 3 or β 3 .
Finding the Correct Values We know that a β€ b . Let's consider the possible values for a and b :
If a = 5 , then b must be greater than or equal to 5 . However, β£ b β£ = 3 , so b can only be 3 or β 3 . Thus, a cannot be 5 .
If a = β 5 , then b must be greater than or equal to β 5 . Since β£ b β£ = 3 , b can be 3 or β 3 . Both 3 and β 3 are greater than β 5 , so a = β 5 is possible.
If b = 3 , then a β€ 3 . Since β£ a β£ = 5 , a can be 5 or β 5 . Since a β€ b , a can be β 5 .
If b = β 3 , then a β€ β 3 . Since β£ a β£ = 5 , a can be 5 or β 5 . Since a β€ b , a can be β 5 .
Determining the Values of a and b Therefore, we must have a = β 5 and b = 3 .
Calculating a+b Now we can find a + b = β 5 + 3 = β 2 .
Final Answer Thus, a + b = β 2 .
Examples
Understanding absolute values and inequalities is crucial in various fields, such as physics and engineering. For example, when analyzing the stability of a structure, engineers need to consider the range of forces acting on it. If the force exceeds a certain threshold, the structure may collapse. By using absolute values and inequalities, engineers can determine the safe operating range and prevent potential disasters. Similarly, in physics, absolute values are used to represent the magnitude of a vector, such as velocity or acceleration, regardless of its direction. Inequalities help define the boundaries within which a physical system operates, ensuring its stability and predictability.
