On the number line shown, is zero halfway between r and s? s is to the right of zero The...

Understanding the Question

We need to determine whether zero is halfway between r and s on a number line.

For zero to be halfway between two points, those points must be equidistant from zero but on opposite sides. This means one point must be the negative of the other (like -3 and +3).

In mathematical terms, we're asking: Is \(\mathrm{r + s = 0}\)? Or equivalently: Is \(\mathrm{s = -r}\)?

This is a yes/no question. We need sufficient information to answer definitively YES (if \(\mathrm{r + s = 0}\)) or NO (if \(\mathrm{r + s ≠ 0}\)).

Analyzing Statement 1

Statement 1: s is to the right of zero (\(\mathrm{s > 0}\)).

This tells us s is positive, but we know nothing about where r is located.

Let's test different scenarios:

• Case 1: If \(\mathrm{r = -5}\) and \(\mathrm{s = 5}\), then \(\mathrm{r + s = 0}\) → YES, zero is halfway between them
• Case 2: If \(\mathrm{r = -3}\) and \(\mathrm{s = 5}\), then \(\mathrm{r + s = 2 ≠ 0}\) → NO, zero is not halfway between them

Since we can get both YES and NO answers depending on where r is placed, Statement 1 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Important: We now forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: The distance between t and r equals the distance between t and -s.

This means t is equidistant from r and -s. In other words, t is the midpoint between r and -s.

But this tells us nothing about the relationship between r and s themselves. Let's test scenarios:

• Scenario 1: \(\mathrm{r = -3, s = 3, t = -3}\)
  - Check: \(\mathrm{t = -3}\) is indeed the midpoint between \(\mathrm{r = -3}\) and \(\mathrm{-s = -3}\) ✓
  - Result: \(\mathrm{r + s = 0}\) → YES, zero is halfway between r and s
• Scenario 2: \(\mathrm{r = 2, s = 2, t = 0}\)
  - Check: \(\mathrm{t = 0}\) is indeed the midpoint between \(\mathrm{r = 2}\) and \(\mathrm{-s = -2}\) ✓
  - Result: \(\mathrm{r + s = 4 ≠ 0}\) → NO, zero is not halfway between r and s

Different scenarios give different answers, so Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Statements

Now we use both statements together:

• From Statement 1: \(\mathrm{s > 0}\) (s is positive)
• From Statement 2: t is equidistant from r and -s

Even with both constraints, we still can't determine whether \(\mathrm{r + s = 0}\).

Let's verify with concrete examples that satisfy both statements:

Example 1: \(\mathrm{r = -1, s = 1, t = -1}\)

• Statement 1: \(\mathrm{s = 1 > 0}\) ✓
• Statement 2: \(\mathrm{t = -1}\) is the midpoint between \(\mathrm{r = -1}\) and \(\mathrm{-s = -1}\) ✓
• Question: \(\mathrm{r + s = 0}\) → YES answer

Example 2: \(\mathrm{r = 3, s = 1, t = 1}\)

• Statement 1: \(\mathrm{s = 1 > 0}\) ✓
• Statement 2: \(\mathrm{t = 1}\) is the midpoint between \(\mathrm{r = 3}\) and \(\mathrm{-s = -1}\) ✓
  (Check: distance from t to r is \(\mathrm{|1 - 3| = 2}\), distance from t to -s is \(\mathrm{|1 - (-1)| = 2}\) ✓)
• Question: \(\mathrm{r + s = 4 ≠ 0}\) → NO answer

Both examples satisfy both statements but give different answers to our question. Therefore, the statements together are NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice C.

The Answer: E

Since neither statement alone nor both statements together provide sufficient information to determine whether zero is halfway between r and s, the answer is E.

Answer Choice E: The statements together are not sufficient.