Exam 2: Linear Models, Equations, and Inequalities

Translate the sentence to an algebraic inequality. -Assume that the mathematical model $\mathrm { C } ( \mathrm { x } ) = 16 \mathrm { x } + 140$ represents the cost $\mathrm { C }$ , in hundreds of dollars, for a certain manufacturer to produce $x$ items. How many items $x$ can be manufactured while keeping costs between between $\$ 526,000$ and $\$ 782,000$ ?

The graph of a certain function $y = f ( x )$ and the zero of that function is given. Using this graph, find a) the $x$ -intercept of the graph of $y = f ( x )$ and b) the solution to the equation $f ( x ) = 0$ . -

Solve the equation for y. - $x - 5 y = 4$

Find the linear function that is the best fit for the given data. Round decimal values to the nearest hundredth, if necessary. - 6 8 20 28 36 2 4 13 20 30

Solve the equation. - $- \frac { 1 } { 2 } ( x - 6 ) - \frac { 1 } { 3 } ( x - 3 ) = x + 1$

Solve the formula for the specified variable. - $\mathrm { A } = \frac { 1 } { 2 } \mathrm {~h} \left( \mathrm {~b} _ { 1 } + \mathrm { b } _ { 2 } \right)$ for $\mathrm { b } _ { 1 }$

Solve the system of equations graphically, if a solution exists. - \[\begin{array} { c } 4 x + y = 6 \\ x + 4 y = 9

Translate the sentence to an algebraic inequality. -Fantastic Flags, Inc. finds that the cost to make $x$ flags is $C = 32 x + 18,695$ , while the revenue produced from them is $R = 36 x$ ( $C$ and $R$ are in dollars). What is the smallest whole number of flags, $x$ , that must be sold for the company to show a profit?

Provide an appropriate response. - $\text { If } \mathrm { b } < 0 \text {, is it true that } \mathrm { b } ^ { 2 } > \mathrm { b } \text { ? Explain. }$

Write the best-fit linear model for the data. -The ages and lengths of several animals of the same species are recorded in the following table: Age (months) Length (inches) 12 9 15 12 17 20 21 21 26 24 28 27 32 35 38 40 41 40 Find and then use a linear model to determine the age of an animal whose length is 30 inches. Round to the nearest year.

Solve the double inequality. - $14 < 5 x + 4 \leq 34$

The graph of a certain function $y = f ( x )$ and the zero of that function is given. Using this graph, find a) the $x$ -intercept of the graph of $y = f ( x )$ and b) the solution to the equation $f ( x ) = 0$ . -

Solve the equation. - $\frac { 1 } { 5 } ( r + 6 ) = \frac { 1 } { 7 } ( r + 8 )$

Provide an appropriate response. -A linear model is obtained for the following set of data. 2 4 6 9 10 12 28 33 39 45 47 52 For what range of x-values would it be reasonable to use the model to predict the y-value corresponding to a given x-value? Why?

Solve the inequality graphically. Give the solution in interval notation. -Use the intersection method to solve $13 x - 4 \leq 12 x - 2$

To find the number of units that gives break-even for the product, solve the equation R $\mathbf { R } = \mathbf { C }$ ound your answer to the nearest whole unit. -Suppose that the total annual consumption of salmon in a certain country is given by $y = 5.59 x + 837.5$ and that the total annual consumption of tuna in this country is given by $y = 1.49 x + 878.5$ , where consumption is measured in millions of pounds and $x$ is the number of years since 2010 . When consumption of salmon reaches consumption of tuna, what is the annual consumption of salmon?

Provide an appropriate response. -A pediatric speech therapist started her own practice in 2004. The table below shows the number of children she treated each year from 2004 to 2012. The data is to be aligned to the number of years past 2000 and fitted with a linear model. Year 2004 2005 2006 2007 2008 2009 2010 2011 2012 Number of Patients 11 15 16 20 23 25 29 34 37 Classify each of the following as discrete or continuous. i) The data in the table ii) A scatter plot made from the data iii) The best-fit linear model obtained for the data iv) The graph of the best-fit linear model obtained for the data

Provide an appropriate response. -A pediatric speech therapist started her own practice in 2004. The function y $y = 3.2 x + 10.53$ odels the number of patients she treated each year, where x is the number of years after 2004. The model was found using data for The years between 2004 and 2012. What does the model estimate as the number of patients in 2014? Round to The nearest whole number. Is this interpolation or extrapolation?

The graph of a certain function $y = f ( x )$ and the zero of that function is given. Using this graph, find a) the $x$ -intercept of the graph of $y = f ( x )$ and b) the solution to the equation $f ( x ) = 0$ . -

Without graphing, determine whether the following data set is exactly linear, approximately linear or nonlinear. - 1 2 3 4 5 9 13 17 21 25