If a, b and c are three real numbers, then which of the following is NOT true ?

Options

A. `|a+b| le|a| + |b|`
B. `|a-b| le |a| + |b|`
C. `|a-b| le |a| - |b|`
D. `|a-c|le |a-b|+|b-c|`

Answer

To determine which of the following statements about the real numbers \( a \), \( b \), and \( c \) is NOT true, we will analyze each option step by step.

### Given Values:
Let:
- \( a = 1 \)
- \( b = -2 \)
- \( c = 0 \)

### Option A:
**Statement:** \( |a + b| \leq |a| + |b| \)

**Calculation:**
- Left side: \( |a + b| = |1 - 2| = | -1 | = 1 \)
- Right side: \( |a| + |b| = |1| + |-2| = 1 + 2 = 3 \)

**Comparison:**
- \( 1 \leq 3 \) is true.

### Option B:
**Statement:** \( |a - b| \leq |a| + |b| \)

**Calculation:**
- Left side: \( |a - b| = |1 - (-2)| = |1 + 2| = |3| = 3 \)
- Right side: \( |a| + |b| = |1| + |-2| = 1 + 2 = 3 \)

**Comparison:**
- \( 3 \leq 3 \) is true.

### Option C:
**Statement:** \( |a - b| \leq |a| - |b| \)

**Calculation:**
- Left side: \( |a - b| = |1 - (-2)| = |3| = 3 \)
- Right side: \( |a| - |b| = |1| - |-2| = 1 - 2 = -1 \)

**Comparison:**
- \( 3 \leq -1 \) is false.

### Option D:
**Statement:** (Assuming a similar form as previous options, but not provided in the transcript)

### Conclusion:
From the analysis, we find that the statement in **Option C** is NOT true.

### Final Answer:
The statement that is NOT true is **C**.

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