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 $34 \%$ 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 $2012 .$ 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 $34 \%$ 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

Systems of Inequalities: Intersection and Union

Systems of inequalities involve combining multiple inequality conditions. The intersection of inequalities uses the 'and' condition, meaning that all conditions must be satisfied simultaneously, which results in the overlap of the solution sets. Conversely, the union uses the 'or' condition, meaning that satisfying any one of the inequalities is sufficient. This concept is essential when evaluating combined criteria in data models.

Solving Linear Inequalities

Solving linear inequalities involves finding the set of values for the variable that make the inequality true. This process is similar to solving linear equations, with extra attention paid to the inequality sign. It requires isolating the variable, and is used in modeling problems to determine when a given condition will be met, such as when a percentage exceeds a certain threshold.

Interpreting Time Variables in Models

When a model uses a variable for time such as x, it is important to recognize what x represents, typically the number of years since a base year. This allows you to accurately determine the actual year corresponding to a specific value of x by adding it to the start year. Understanding this concept is crucial when converting model outputs into real-world time frames.

Linear Function Models

Linear function models are equations of the form y = mx + b that represent a constant rate of change. In the context of data modeling, these functions are used to project trends over time based on an initial value (the y-intercept) and a consistent change quantified by the slope. They provide a simplified representation of dynamic scenarios, such as increases or decreases in percentages over the years.

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