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 $2012 .$ Also shown is the percentage of households in which a person of faith is married to someone with no religion. GRAPH CAN'T COPY. The formula $$I=\frac{1}{4} x+26$$ 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 $1988 .$ Use these models to solve Exercises $107-108$. 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 $2012 .$ Also shown is the percentage of households in which a person of faith is married to someone with no religion.
GRAPH CAN'T COPY.
The formula
$$I=\frac{1}{4} x+26$$
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 $1988 .$ Use these models to solve Exercises $107-108$.
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 Models

A linear model is used to represent a relationship between two variables with a constant rate of change. In this context, the percentages of certain types of marriages are modeled as linear functions of time, where the independent variable represents the number of years since a base year. This helps to predict future percentages based on the observed trend over time.

Solving Linear Inequalities

Linear inequalities involve finding the range of values for the variable that makes the inequality true. By setting the linear model expressions greater than the given thresholds, one can solve for the variable to determine the time intervals when the percentages exceed those thresholds.

Mapping Variables to Real-World Time

The variable in the model represents the number of years elapsed since a base year. Once the inequality is solved for the variable, the result must be translated into calendar years by adding the base year, thus connecting the mathematical solution to the real-world context.

Intersection and Union of Conditions

When multiple conditions or events are considered simultaneously, 'AND' (intersection) is used to find the period during which all conditions are met, while 'OR' (union) is used to determine when at least one of the conditions holds. This logical framework is important to combine different scenarios based on the results of the individual inequalities.

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