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 CANNOT 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. 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 CANNOT 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.
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

Solving Linear Inequalities

Solving linear inequalities involves manipulating the inequality by performing arithmetic operations to isolate the variable. This process determines the range of input values that cause the function to exceed, fall below, or equal a certain threshold, which is essential in comparing modeled predictions to specified criteria.

Intersection and Union of Solution Sets

When multiple conditions are presented, interpreting their intersection (logical 'and') helps identify the scenarios where all conditions are satisfied simultaneously, whereas the union (logical 'or') combines all scenarios where at least one condition is met. This is critical for analyzing combined events using separate linear models.

Variable Interpretation and Domain Analysis

Understanding the role of variables in a model is key; typically, the variable represents an independent factor such as time, while its domain must adhere to the context of the problem. Correct interpretation and adjustment of the base year or starting point ensure that the model's outputs are meaningful and correctly applied to real-world scenarios.

Linear Functions

Linear functions model relationships where one variable changes at a constant rate with respect to another. They are used to predict values with a straightforward formula consisting of a slope and an intercept, making them ideal for tracking changes over time in various contexts.

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