In more U.S. marriages, spouses have different faiths. The...

In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and $2008 .$ Also shown is the percentage of households in which a person of faith is married to someone with no religion. (GRAPH CANT COPY) The formula $$1-\frac{1}{2} x+2=$$ models the percentage of U.S. households with an interfaith marriage, $I, x$ years after $1988 .$ The formula $$N=\frac{1}{4} x+6$$ models the percentage of U.S. households in which a person of faith is married to someone with no religion, $N, x$ years after I988. Use these models to solve. a. In which years will more than $33 \%$ of U.S. households have an interfaith marriage? b. In which years will more than $14 \%$ of U.S. households have a person of faith married to someone with no religion? c. Based on your answers to parts (a) and (b), in which years will more than $33 \%$ of households have an interfaith marriage and more than $14 \%$ have a faith/no religion marriage? d. Based on your answers to parts (a) and (b), in which years will more than $33 \%$ of households have an interfaith marriage or more than $14 \%$ have a faith/no religion marriage?

In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and $2008 .$ Also shown is the percentage of
households in which a person of faith is married to someone with no religion.
(GRAPH CANT COPY)
The formula
$$1-\frac{1}{2} x+2=$$
models the percentage of U.S. households with an interfaith
marriage, $I, x$ years after $1988 .$ The formula
$$N=\frac{1}{4} x+6$$
models the percentage of U.S. households in which a person of faith is married to someone with no religion, $N, x$ years after I988. Use these models to solve.
a. In which years will more than $33 \%$ of U.S. households have an interfaith marriage?
b. In which years will more than $14 \%$ of U.S. households have a person of faith married to someone with no religion?
c. Based on your answers to parts (a) and (b), in which years will more than $33 \%$ of households have an interfaith marriage and more than $14 \%$ have a faith/no religion marriage?
d. Based on your answers to parts (a) and (b), in which years will more than $33 \%$ of households have an interfaith marriage or more than $14 \%$ have a faith/no religion marriage?

Key Concepts

Linear Functions as Models

In many real-world applications, relationships between two variables are expressed using linear functions. These functions are characterized by a constant rate of change, represented by the slope, and an initial value, represented by the y-intercept. In contexts such as demographic trends or percentages in populations, linear models provide a simple, yet effective, way to predict future outcomes or interpret past data over time.

Time Variables and Model Baselines

When using mathematical models to represent changes over time, it is common to define a time variable relative to a baseline year. This approach simplifies the model and interpretation by allowing the independent variable to represent the number of years elapsed since the base date. Converting between the model's time variable and actual calendar years is an essential step in applying the model to real-world scenarios.

Solving Linear Inequalities

Linear inequalities involve finding the set of values for the independent variable that satisfy a given condition where the linear function is greater or less than a specified threshold. This process usually involves isolating the variable on one side of the inequality and determining the corresponding interval. The solutions to these inequalities then provide ranges for predictions or conditions within the modeled scenario.

Intersection and Union of Inequality Solutions

In many problems, multiple conditions may need to be satisfied simultaneously or individually. The intersection of solutions refers to finding the common values that satisfy all given inequalities, representing an 'AND' condition, while the union of solutions includes all values that satisfy at least one of the inequalities, representing an 'OR' condition. Understanding these set operations is essential for correctly interpreting combined criteria in mathematical models.

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