Is |a-c| + |a| = |c|? ab > bc \(\mathrm{ab}...

Understanding the Question

We need to determine whether \(|\mathrm{a}-\mathrm{c}| + |\mathrm{a}| = |\mathrm{c}|\). This is a yes/no question about absolute values.

What This Equation Means

Let's visualize this on a number line:

• \(|\mathrm{a}-\mathrm{c}|\) = distance between points a and c
• \(|\mathrm{a}|\) = distance from a to 0
• \(|\mathrm{c}|\) = distance from c to 0

The equation asks: Does traveling from a to c, then from a to 0, equal the direct distance from c to 0?

Key Insight

This equation holds true when and only when 0 lies between a and c on the number line.

Here's why: When you go from a to c and pass through 0, the sum of distances equals the total distance. But if 0 is outside the interval between a and c, you're taking an indirect path that's longer than the direct distance \(|\mathrm{c}|\).

For sufficiency: We need to definitively know whether 0 lies between a and c.

Analyzing Statement 1

Statement 1: \(\mathrm{ab} > \mathrm{bc}\)

This can be rewritten as \(\mathrm{b}(\mathrm{a}-\mathrm{c}) > 0\), meaning b and (a-c) have the same sign.

What We Can Determine

If \(\mathrm{b} > 0\): then \(\mathrm{a}-\mathrm{c} > 0\), so \(\mathrm{a} > \mathrm{c}\)
If \(\mathrm{b} < 0\): then \(\mathrm{a}-\mathrm{c} < 0\), so \(\mathrm{a} < \mathrm{c}\)

This tells us the relative positions of a and c, but nothing about where 0 sits.

Testing Different Scenarios

Scenario 1: Let \(\mathrm{a} = 2\), \(\mathrm{b} = 1\), \(\mathrm{c} = 1\)

• Verify \(\mathrm{ab} > \mathrm{bc}\): \(2 > 1\) ✓
• Position check: 0 is outside the interval [1,2]
• Check equation: \(|2-1| + |2| = 1 + 2 = 3\), while \(|1| = 1\)
• Result: \(3 ≠ 1\), equation is FALSE

Scenario 2: Let \(\mathrm{a} = 0\), \(\mathrm{b} = 1\), \(\mathrm{c} = -1\)

• Verify \(\mathrm{ab} > \mathrm{bc}\): \(0 > -1\) ✓
• Position check: \(0 = \mathrm{a}\), so it's at the endpoint
• Check equation: \(|0-(-1)| + |0| = 1 + 0 = 1\), while \(|-1| = 1\)
• Result: \(1 = 1\), equation is TRUE

Different scenarios give different answers.

Statement 1 is NOT sufficient. [STOP - Not Sufficient!]

This eliminates choices A and D.

Analyzing Statement 2

Now we forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: \(\mathrm{ab} < 0\)

This means a and b have opposite signs. But this tells us nothing about c or where c sits relative to 0 and a.

Testing Different Scenarios

Scenario 1: Let \(\mathrm{a} = 2\), \(\mathrm{b} = -1\), \(\mathrm{c} = 1\)

• Verify \(\mathrm{ab} < 0\): \((2)(-1) = -2 < 0\) ✓
• Position check: 0 is outside the interval between a and c
• Check equation: \(|2-1| + |2| = 1 + 2 = 3\), while \(|1| = 1\)
• Result: \(3 ≠ 1\), equation is FALSE

Scenario 2: Let \(\mathrm{a} = 1\), \(\mathrm{b} = -1\), \(\mathrm{c} = -3\)

• Verify \(\mathrm{ab} < 0\): \((1)(-1) = -1 < 0\) ✓
• Position check: 0 lies between \(\mathrm{c} = -3\) and \(\mathrm{a} = 1\)
• Check equation: \(|1-(-3)| + |1| = 4 + 1 = 5\), while \(|-3| = 3\)
• Result: \(5 ≠ 3\), equation is FALSE

Wait, this seems wrong. Let me reconsider...

Actually, for the equation to be TRUE when 0 is between a and c, we need a specific arrangement. Let me try:

Revised Scenario 2: Let \(\mathrm{a} = -2\), \(\mathrm{b} = 1\), \(\mathrm{c} = 3\)

• Verify \(\mathrm{ab} < 0\): \((-2)(1) = -2 < 0\) ✓
• Position check: 0 lies between \(\mathrm{a} = -2\) and \(\mathrm{c} = 3\)
• Check equation: \(|-2-3| + |-2| = 5 + 2 = 7\), while \(|3| = 3\)
• Result: \(7 ≠ 3\), equation is FALSE

The issue is that for \(|\mathrm{a}-\mathrm{c}| + |\mathrm{a}| = |\mathrm{c}|\) to be true, we need the specific case where a and c are on opposite sides of 0, AND going from a to c passes through 0 in a way that makes the sum equal to \(|\mathrm{c}|\).

Better Scenario 2: Let \(\mathrm{a} = -1\), \(\mathrm{b} = 1\), \(\mathrm{c} = 2\)

• Verify \(\mathrm{ab} < 0\): \((-1)(1) = -1 < 0\) ✓
• Check equation: \(|-1-2| + |-1| = 3 + 1 = 4\), while \(|2| = 2\)
• Result: \(4 ≠ 2\), equation is FALSE

Correct Scenario 2: Let \(\mathrm{a} = 1\), \(\mathrm{b} = -1\), \(\mathrm{c} = 2\)

• Verify \(\mathrm{ab} < 0\): \((1)(-1) = -1 < 0\) ✓
• Check equation: \(|1-2| + |1| = 1 + 1 = 2\), while \(|2| = 2\)
• Result: \(2 = 2\), equation is TRUE

Different scenarios give different answers.

Statement 2 is NOT sufficient. [STOP - Not Sufficient!]

This eliminates choice B (and D was already eliminated).

Combining Statements

Now we use both statements together:

• From Statement 1: \(\mathrm{ab} > \mathrm{bc}\), which means \(\mathrm{b}(\mathrm{a}-\mathrm{c}) > 0\)
• From Statement 2: \(\mathrm{ab} < 0\), so a and b have opposite signs

Case Analysis

Case 1: If \(\mathrm{a} > 0\) and \(\mathrm{b} < 0\)

• We have \(\mathrm{ab} < 0\) ✓ and need \(\mathrm{ab} > \mathrm{bc}\)
• Since \(\mathrm{b} < 0\), we're comparing two negative values: ab (negative) > bc
• Dividing by \(\mathrm{b} < 0\) flips the inequality: \(\mathrm{a} < \mathrm{c}\)
• Final arrangement: \(0 < \mathrm{a} < \mathrm{c}\)

Case 2: If \(\mathrm{a} < 0\) and \(\mathrm{b} > 0\)

• We have \(\mathrm{ab} < 0\) ✓ and need \(\mathrm{ab} > \mathrm{bc}\)
• ab is negative while b is positive, so bc must be even more negative
• This means \(\mathrm{c} < 0\) (since \(\mathrm{b} > 0\) and bc is very negative)
• With \(\mathrm{b} > 0\): \(\mathrm{ab} > \mathrm{bc}\) gives us \(\mathrm{a} > \mathrm{c}\)
• Final arrangement: \(\mathrm{c} < \mathrm{a} < 0\)

The Crucial Observation

In both cases, 0 lies between a and c!

• Case 1: a and c are both positive with \(0 < \mathrm{a} < \mathrm{c}\)
• Case 2: a and c are both negative with \(\mathrm{c} < \mathrm{a} < 0\)

Verification

When 0 lies between a and c in this way:

• Case 1 (\(0 < \mathrm{a} < \mathrm{c}\)): \(|\mathrm{a}-\mathrm{c}| + |\mathrm{a}| = (\mathrm{c}-\mathrm{a}) + \mathrm{a} = \mathrm{c} = |\mathrm{c}|\) ✓
• Case 2 (\(\mathrm{c} < \mathrm{a} < 0\)): \(|\mathrm{a}-\mathrm{c}| + |\mathrm{a}| = (\mathrm{a}-\mathrm{c}) + (-\mathrm{a}) = -\mathrm{c} = |\mathrm{c}|\) ✓

The equation is TRUE in both cases, so we can definitively answer YES.

Both statements together are SUFFICIENT. [STOP - Sufficient!]

This eliminates choice E.

The Answer: C

Both statements together guarantee that 0 must lie between a and c, which makes the equation \(|\mathrm{a}-\mathrm{c}| + |\mathrm{a}| = |\mathrm{c}|\) true.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."