To determine the two unique numbers plotted by Bernita and Derek that have the same absolute value and whose absolute values sum to 150, let’s follow a step-by-step approach:
1. Understand the problem statement:
- We have two numbers with the same absolute value.
- Their absolute values sum to 150.
- The numbers are unique.
2. Define the variables:
- Let the two numbers be [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
- Since they have the same absolute value, we can write them as [tex]\( |a| \)[/tex] and [tex]\( |b| \)[/tex].
- Because they are unique but have the same absolute value, one will be positive and the other negative, so let's say [tex]\( a = x \)[/tex] and [tex]\( b = -x \)[/tex] where [tex]\( |x| \)[/tex] is the absolute value of the numbers.
3. Set up the equation:
- According to the problem, the sum of their absolute values is:
[tex]\[
|a| + |b| = 150
\][/tex]
- Substituting [tex]\( |a| = |x| \)[/tex] and [tex]\( |b| = |-x| \)[/tex] into the equation:
[tex]\[
|x| + |-x| = 150
\][/tex]
4. Simplify the equation:
- Since the absolute value of [tex]\( x \)[/tex] and [tex]\( -x \)[/tex] is the same:
[tex]\[
|x| + |x| = 150
\][/tex]
- This simplifies to:
[tex]\[
2|x| = 150
\][/tex]
5. Solve for [tex]\( |x| \)[/tex]:
- Dividing both sides by 2:
[tex]\[
|x| = \frac{150}{2} = 75
\][/tex]
6. Determine the numbers:
- We know the absolute value of [tex]\( x \)[/tex] is 75, so:
[tex]\[
x = 75 \quad \text{and} \quad -x = -75
\][/tex]
Therefore, the two unique numbers plotted by Bernita and Derek that satisfy all the conditions given are:
[tex]\[
75 \quad \text{and} \quad -75
\][/tex]
And the correct answer is:
-75 and 75
