For any two real numbers a and b, \(\sqrt {{{\left( {a - b} \right)}^2}} + \sqrt {{{\left( {b - a} \right)}^2}} \) is
positive only if a ≠ b
Understanding the Expression with Real Numbers
We are asked to evaluate the expression \( \sqrt {{{\left( {a - b} \right)}^2}} + \sqrt {{{\left( {b - a} \right)}^2}} \) for any two real numbers \(a\) and \(b\). We need to determine the nature of its value.
Applying the Square Root Property
A key property involving square roots and squares is that for any real number \(x\), the square root of \(x\) squared is the absolute value of \(x\). This is written as \( \sqrt{x^2} = |x| \).
Let's apply this property to the terms in our expression:
- For the first term, let \(x = a - b\). Then \( \sqrt{{{\left( {a - b} \right)}^2}} = |a - b| \).
- For the second term, let \(x = b - a\). Then \( \sqrt{{{\left( {b - a} \right)}^2}} = |b - a| \).
So, the original expression can be rewritten as \( |a - b| + |b - a| \).
Simplifying Using Absolute Value Properties
Next, we use another property of absolute value: for any real number \(y\), \(|-y| = |y|\). Notice that \(b - a\) is the negative of \(a - b\), i.e., \(b - a = -(a - b)\).
Using the property \(|-y| = |y|\), we have \(|b - a| = |-(a - b)| = |a - b|\).
Now substitute this back into our simplified expression:
\( |a - b| + |b - a| = |a - b| + |a - b| = 2|a - b| \).
The expression simplifies to \( 2|a - b| \).
Analyzing the Value Based on the Relationship Between a and b
The value of \( 2|a - b| \) depends on the value of \(|a - b|\), which in turn depends on the relationship between \(a\) and \(b\).
- Case 1: When \(a = b\)
If \(a = b\), then \(a - b = 0\). The expression becomes \( 2|a - b| = 2|0| = 2 \times 0 = 0 \). - Case 2: When \(a \ne b\)
If \(a \ne b\), then \(a - b\) is a non-zero real number. The absolute value of any non-zero real number is always positive, i.e., \(|a - b| > 0\).
The expression becomes \( 2|a - b| \). Since \(|a - b|\) is positive, \( 2|a - b| \) will be \(2\) times a positive number, which is always a positive number. So, \( 2|a - b| > 0 \).
From this analysis, we see that the expression \( \sqrt {{{\left( {a - b} \right)}^2}} + \sqrt {{{\left( {b - a} \right)}^2}} \) is equal to \( 2|a - b| \). This value is \(0\) if \(a = b\) and positive if \(a \ne b\).
This means the expression is positive if and only if \(a \ne b\).
Evaluating the Given Options
Let's look at the given options in light of our findings:
- Option 1: always zero
This is false. The expression is zero only when \(a = b\), not always. - Option 2: never zero
This is false. The expression is zero when \(a = b\). - Option 3: positive only if a \(\ne\) b
This statement means "If the expression is positive, then \(a \ne b\)". Our analysis showed that the expression is positive exactly when \(a \ne b\). This implies both "If \(a \ne b\), the expression is positive" and "If the expression is positive, then \(a \ne b\)". Therefore, this statement accurately describes the condition for the expression to be positive. - Option 4: positive if and only if a > b
This is false. The expression is positive if \(a \ne b\), which includes cases where \(a < b\) (e.g., \(a=1, b=2\), expression is \(2|1-2| = 2\)).
Final Answer Selection
Based on our detailed simplification and analysis of the cases for real numbers \(a\) and \(b\), the expression \( \sqrt {{{\left( {a - b} \right)}^2}} + \sqrt {{{\left( {b - a} \right)}^2}} \) is equal to \( 2|a - b| \). This value is positive precisely when \(a \ne b\). Option 3 correctly states this condition.
Revision Table: Key Concepts for Square Roots and Absolute Value
| Concept | Property | Explanation |
|---|---|---|
| Square root of a square | \( \sqrt{x^2} = |x| \) | For any real number \(x\), the square root of its square is its absolute value. It is always non-negative. |
| Absolute value definition | \( |x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases} \) | The absolute value of \(x\) is its distance from zero on the number line. It is always non-negative. |
| Absolute value of negative | \( |-y| = |y| \) | The absolute value of a number is the same as the absolute value of its negative. |
Additional Information: Properties of Absolute Value for Real Numbers
The absolute value function \(|x|\) is fundamental in algebra and analysis. Here are some important properties for real numbers \(x\) and \(y\):
- Non-negativity: \(|x| \ge 0\)
- Identity of indiscernibles: \(|x| = 0 \iff x = 0\)
- Multiplicativity: \(|xy| = |x||y|\)
- Triangle inequality: \(|x + y| \le |x| + |y|\)
- Reverse triangle inequality: \(|x - y| \ge ||x| - |y||\)
In this problem, we used \( \sqrt{x^2} = |x| \) and \( |-y| = |y| \). The fact that \( |a - b| = 0 \iff a - b = 0 \iff a = b \) is derived from the identity of indiscernibles, which helps us determine when the expression is zero versus positive.
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