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 $33 \%$ 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 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 $33 \%$ 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

Combining Inequalities with Logical Operators

When dealing with multiple conditions simultaneously, it is essential to understand logical operators such as 'and' (intersection) and 'or' (union). The 'and' operator requires that all conditions be met simultaneously, while the 'or' operator requires that at least one of the conditions is met. This distinction is crucial when determining ranges that satisfy compound conditions in mathematical models.

Linear Functions

Linear functions are mathematical models that describe a constant rate of change between two variables. They are often used to represent relationships in real-world scenarios where there is a steady increase or decrease over time. The general form of a linear function is y = mx + b, where m represents the rate of change and b is the y-intercept, indicating the value when x is zero.

Solving Linear Inequalities

Solving linear inequalities involves finding the range of values for the variable that satisfy an inequality expressed by a linear function. This typically requires isolating the variable on one side of the inequality sign, similar to solving linear equations, but with the added rule that when both sides of an inequality are multiplied or divided by a negative number, the inequality sign must be reversed.

Translating Model Variables to Real-World Contexts

In many applications, variables in mathematical models (such as the x in a linear function) represent real-world quantities like time. Interpreting the model correctly requires understanding how to convert between the mathematical variable and the actual context, such as determining the specific calendar years that correspond to given values of x after an initial point.

Systems of Inequalities

Systems of inequalities involve solving more than one inequality simultaneously. This typically means finding the set of variable values that satisfy all individual inequalities (using the 'and' operator) or any of them (using the 'or' operator). Understanding how to graph these systems or combine their solutions mathematically allows for solving complex real-world problems that involve multiple constraints.

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