10. In 2006 the median yearly family income was about \( \$ 48,200 \) per year. Suppose the average annual rate of change since then is \( \$ 1240 \) per year. a. Write and graph an inequality for the annual family incomes \( y \) that are less than the median for years after 2006 . b. Deromine whether each of the following points is part of the solution set. \( \begin{array}{llll}(2,51,000) & (8,69,200) & (5,50,000) & (10,61,000)\end{array} \)

To start off, the median family income for 2006 is given as \( \$ 48,200 \). With an average annual increase of \( \$ 1,240 \), we can express the income after \( t \) years since 2006 as: \[ y < 48,200 + 1,240t \] So, the inequality for the annual family incomes \( y \) that are less than the median over the years post-2006 can be written as: \[ y < 48,200 + 1,240(t) \] Next, let’s evaluate the specified points \( (t, y) \): - For \( (2, 51,000) \): \[ 51,000 < 48,200 + 1,240(2) \\ 51,000 < 48,200 + 2,480 \\ 51,000 < 50,680 \quad \text{(false)} \] - For \( (8, 69,200) \): \[ 69,200 < 48,200 + 1,240(8) \\ 69,200 < 48,200 + 9,920 \\ 69,200 < 58,120 \quad \text{(false)} \] - For \( (5, 50,000) \): \[ 50,000 < 48,200 + 1,240(5) \\ 50,000 < 48,200 + 6,200 \\ 50,000 < 54,400 \quad \text{(true)} \] - For \( (10, 61,000) \): \[ 61,000 < 48,200 + 1,240(10) \\ 61,000 < 48,200 + 12,400 \\ 61,000 < 60,600 \quad \text{(false)} \] In summary, out of the given points, only \( (5, 50,000) \) falls within the defined income inequality. For an extra sprinkle of intrigue: Did you know the economic landscape has changed dramatically since 2006? That time marked the beginning of new economic and technological advancements, influencing family incomes significantly later on! Moreover, trends in income inequality have been a hot topic ever since, affecting how families budget and manage their finances. More insights can be discovered in reports compiled by the U.S. Census Bureau, showcasing shifts in median family incomes over the years!