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 $34 \%$ of U.S. households have an interfaith marriage? b. In which years will more than $15 \%$ 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 $34 \%$ of households have an interfaith marriage and more than $15 \%$ have a faith/no religion marriage? d. Based on your answers to parts (a) and (b), in which years will more than $34 \%$ of households have an interfaith marriage or more than $15 \%$ 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 $34 \%$ of U.S. households have an interfaith marriage?
b. In which years will more than $15 \%$ 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 $34 \%$ of households have an interfaith marriage and more than $15 \%$ have a faith/no religion marriage?
d. Based on your answers to parts (a) and (b), in which years will more than $34 \%$ of households have an interfaith marriage or more than $15 \%$ have a faith/no religion marriage?

Key Concepts

Linear Functions

Linear functions describe relationships with a constant rate of change and are expressed in the form y = mx + b. They are used in modeling by providing a way to predict an outcome based on an input variable, such as time, where each unit change in time corresponds to a fixed change in the percentage value.

Mathematical Modeling

Mathematical modeling involves creating equations or formulas that represent real-world scenarios. In this context, linear functions are used to model the percentages of households for different types of marriages over time, helping to predict future trends based on past data.

Linear Inequalities

Linear inequalities are used to determine the range of input values (often time) for which the associated output (such as a percentage) meets a specified condition, like exceeding a particular threshold. The process involves solving the inequality to find the time interval that satisfies the condition.

Time Variables in Models

Time variables in mathematical models indicate the passage of time from a specific starting point (base year). Understanding how to incorporate and interpret time as a variable is crucial for accurately mapping the mathematical solution back into calendar years.

Logical Connectives in Combined Conditions

Logical connectives such as 'and' and 'or' are used to combine multiple criteria. 'And' requires all conditions to be satisfied simultaneously, while 'or' requires at least one condition to hold. This concept is essential when determining periods that meet either one or multiple conditions concurrently.

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