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Question

Part 1
The 20062006 per capita income for a particular country was ​$38 comma 51138,511. The following table shows the per capita income for 20062006 for selected states.
State
Per Capita Income
State A
32 comma 50432,504
State B
41 comma 04241,042
State C
33 comma 45733,457
State D
35 comma 02335,023
State E
47 comma 02447,024
State F
34 comma 38934,389
State G
49 comma 19449,194
State H
34 comma 88134,881
State I
31 comma 02031,020
State J
36 comma 68536,685
Answer parts a and b.
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Question content area right
Part 1
a. If a is the per capita income for a​ state, then for which states is StartAbsoluteValue a minus 38 comma 511 EndAbsoluteValue less than 3000a−38,511<3000​? Choose the correct answer below.

Answer 1

To find out which states satisfy the condition \( |a - 38,511| < 3,000 \), we need to determine the range of per capita income values that will satisfy this inequality.

1. Start by solving the inequality: \[ |a - 38,511| < 3,000 \] This inequality implies: \[ -3,000 < a - 38,511 < 3,000 \]

2. Adding \( 38,511 \) to each part of the compound inequality: \[ 38,511 - 3,000 < a < 38,511 + 3,000 \] \[ 35,511 < a < 41,511 \]

Now, we can compare this range with the per capita incomes of the selected states:

• State A: \( 32,504 \)
• State B: \( 41,042 \)
• State C: \( 33,457 \)
• State D: \( 35,023 \)
• State E: \( 47,024 \)
• State F: \( 34,389 \)
• State G: \( 49,194 \)
• State H: \( 34,881 \)
• State I: \( 31,020 \)
• State J: \( 36,685 \)

3. Identify the states which fall within the range \( 35,511 < a < 41,511 \):

• State D: \( 35,023 \) (not included; less than 35,511)
• State B: \( 41,042 \) (included; less than 41,511)
• State J: \( 36,685 \) (included; greater than 35,511 and less than 41,511)

Final answer: The states that satisfy the condition \( |a - 38,511| < 3,000 \) are:

• State B: \( 41,042 \)
• State J: \( 36,685 \)

Answer 2

For which states is StartAbsoluteValue a minus 38 comma 511 EndAbsoluteValue greater than 10 comma 000a−38,511>10,000​?
Choose the correct answer below.
A.
State D
B.
State​ E, F,​ G, H
C.
State​ I, J, A
D.State Upper G
State Upper GState G

Answer 3

To find out which states satisfy the condition \( |a - 38,511| > 10,000 \), we need to determine the range of per capita income values that will satisfy this inequality.

1. Start by solving the inequality: \[ |a - 38,511| > 10,000 \] This inequality implies: \[ a - 38,511 > 10,000 \quad \text{or} \quad a - 38,511 < -10,000 \]

2. Solving the two inequalities:

   • For \( a - 38,511 > 10,000 \): \[ a > 38,511 + 10,000 = 48,511 \]

   • For \( a - 38,511 < -10,000 \): \[ a < 38,511 - 10,000 = 28,511 \]

This means the states that satisfy \( |a - 38,511| > 10,000 \) must either have a per capita income less than \( 28,511 \) or greater than \( 48,511 \).

3. Now, we can compare this range with the per capita incomes of the selected states:

• State A: \( 32,504 \)
• State B: \( 41,042 \)
• State C: \( 33,457 \)
• State D: \( 35,023 \)
• State E: \( 47,024 \)
• State F: \( 34,389 \)
• State G: \( 49,194 \)
• State H: \( 34,881 \)
• State I: \( 31,020 \)
• State J: \( 36,685 \)

4. Identify the states which fall within the ranges determined:
   • \( a < 28,511 \): No states fall into this range since all listed values are greater than \( 28,511 \).
   • \( a > 48,511 \): Only State G falls into this range (with \( 49,194 \)).

Final answer: The states that satisfy the condition \( |a - 38,511| > 10,000 \) are:

• State G.