Chapter 8

Solve the inequality and graph the solution on a real number line. \(-11-6 x \geq 33\)

Write an equation of the line passing through the two points. Use the slope- intercept form, if possible. If not possible, explain why. $$(6,3),(10,3)$$

Let matrices \(A,B,C,\) and \(D\) be of dimensions \(2 \times 3,2 \times 3,3 \times 2,\) and \(2 \times 2\) respectively. Determine whether the matrices are of proper dimension to perform the operation(s). If so, give the dimension of the answer. $$C B-D$$

Fitting a Parabola To find the least squares regression parabola \(y=a x^{2}+b x+c\) for a set of points \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) you can solve the following system of linear equations for \(a, b,\) and \(c\) $$\left\\{\begin{aligned} n c+\left(\sum_{i=1}^{n} x_{i}\right) b+\left(\sum_{i=1}^{n} x_{i}^{2}\right) a &=\sum_{i=1}^{n} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}\right) c+\left(\sum_{i=1}^{n} x_{i}^{2}\right) b+\left(\sum_{i=1}^{n} x_{i}^{3}\right) a &=\sum_{i=1}^{n} x_{i} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}^{2}\right) c+\left(\sum_{i=1}^{n} x_{i}^{3}\right) b+\left(\sum_{i=1}^{n} x_{i}^{4}\right) a &=\sum_{i=1}^{n} x_{i}^{2} y_{i} \end{aligned}\right.$$ The sums have been evaluated. Solve the given system for \(a, b,\) and \(c\) to find the least squares regression parabola for the points. Use a graphing utility to confirm the result. $$\left\\{\begin{aligned} 4 c+6 b+14 a &=25 \\ 6 c+14 b+36 a &=21 \\ 14 c+36 b+98 a &=33 \end{aligned}\right.$$

Determine whether the statement is true or false. Justify your answer. You cannot write an augmented matrix for a dependent system of linear equations in reduced row-echelon form.

Solve the inequality and graph the solution on a real number line. \(-6 \leq 3 x-10<6\)

Write an equation of the line passing through the two points. Use the slope- intercept form, if possible. If not possible, explain why. $$(4,-2),(4,5)$$

Let matrices \(A,B,C,\) and \(D\) be of dimensions \(2 \times 3,2 \times 3,3 \times 2,\) and \(2 \times 2\) respectively. Determine whether the matrices are of proper dimension to perform the operation(s). If so, give the dimension of the answer. $$D(A-3 B)$$

During the testing of a new automobile braking system, the speeds \(x\) (in miles per hour) and the stopping distances \(y\) (in feet) were recorded in the table. $$\begin{array}{|c|c|} \hline \text { Speed, } x & \text { Stopping distance, } y \\ \hline 30 & 55 \\ \hline 40 & 105 \\ \hline 50 & 188 \\ \hline \end{array}$$ (a) Use the data to create a system of linear equations. Then find the least squares regression parabola for the data by solving the system. (b) Use a graphing utility to graph the parabola and the data in the same viewing window. (c) Use the model to estimate the stopping distance for a speed of 70 miles per hour.

Think About It The augmented matrix represents a system of linear equations (in the variables \(x, y,\) and \(z\) ) that has been reduced using Gauss-Jordan elimination. Write a system of three equations in three variables with nonzero coefficients that is represented by the reduced matrix. (There are many correct answers.) $$\left[\begin{array}{llllr} 1 & 0 & 3 & \vdots & -2 \\ 0 & 1 & 4 & \vdots & 1 \\ 0 & 0 & 0 & \vdots & 0 \end{array}\right]$$

Solve the inequality and graph the solution on a real number line. \(|x-8|<10\)

Write an equation of the line passing through the two points. Use the slope- intercept form, if possible. If not possible, explain why. $$\left(\frac{3}{5}, 0\right),(4,6)$$

Let matrices \(A,B,C,\) and \(D\) be of dimensions \(2 \times 3,2 \times 3,3 \times 2,\) and \(2 \times 2\) respectively. Determine whether the matrices are of proper dimension to perform the operation(s). If so, give the dimension of the answer. $$C(A+2 B)$$

A wildlife management team studied the reproduction rates of deer in three five-acre tracts of a wildlife preserve. In each tract, the number of females \(x\) and the percent of females \(y\) that had offspring the following year were recorded. The results are shown in the table. $$\begin{array}{|c|c|} \hline \text { Number, } x & \text { Percent, } y \\ \hline 120 & 68 \\ \hline 140 & 55 \\ \hline 160 & 30 \\ \hline \end{array}$$ (a) Use the data to create a system of linear equations. Then find the least squares regression parabola for the data by solving the system. (b) Use a graphing utility to graph the parabola and the data in the same viewing window. (c) Use the model to predict the percent of females that had offspring when there were 170 females.

Determine whether the matrix below is in row-echelon form, reduced row-echelon form, or neither when it satisfies the given conditions. $$\left[\begin{array}{ll} 1 & b \\ c & 1 \end{array}\right]$$ (a) \(b=0, c=0\) (b) \(b \neq 0, c=0\) (c) \(b=0, c \neq 0\) (d) \(b \neq 0, c \neq 0\)

Solve the inequality and graph the solution on a real number line. \(|x+10| \geq-3\)

Find the domain of the function and identify any horizontal or vertical asymptotes. $$f(x)=\frac{5}{x-6}$$

Use the matrices \(A=\left[\begin{array}{rr}2 & -1 \\ 1 & 3\end{array}\right] \quad\) and $\quad B=\left[\begin{array}{rr}-1 & 1 \\ 0 & -2\end{array}\right].$$$\text { Show that }(A+B)^{2} \neq A^{2}+2 A B+B^{2}$$

Environment The predicted cost \(C\) (in thousands of dollars) for a company to remove \(p \%\) of a chemical from its wastewater is given by the model $$C=\frac{120 p}{10,000-p^{2}}, \quad 0 \leq p < 100$$ Write the partial fraction decomposition of the rational function. Verify your result by using the table feature of a graphing utility to create a table comparing the original function with the partial fractions.

Can a \(2 \times 4\) augmented matrix whose entries are all nonzero real numbers represent an independent system of linear equations? Explain.

Solve the inequality and graph the solution on a real number line. \(2 x^{2}+3 x-35<0\)

Find the domain of the function and identify any horizontal or vertical asymptotes. $$f(x)=\frac{2 x-7}{3 x+2}$$

Use the matrices \(A=\left[\begin{array}{rr}2 & -1 \\ 1 & 3\end{array}\right] \quad\) and $\quad B=\left[\begin{array}{rr}-1 & 1 \\ 0 & -2\end{array}\right].$$$\text { Show that }(A-B)^{2} \neq A^{2}-2 A B+B^{2}$$

Thermodynamics The magnitude of the range \(R\) of exhaust temperatures (in degrees Fahrenheit) in an experimental diesel engine is approximated by the model $$R=\frac{2000(4-3 x)}{(11-7 x)(7-4 x)}, \quad 0 \leq x \leq 1$$ where \(x\) is the relative load (in foot-pounds). (a) Write the partial fraction decomposition of the rational function. (b) The decomposition in part (a) is the difference of two fractions. The absolute values of the terms give the expected maximum and minimum temperatures of the exhaust gases. Use a graphing utility to graph each term.

Solve the inequality and graph the solution on a real number line. \(3 x^{2}+12 x>0\)

Find the domain of the function and identify any horizontal or vertical asymptotes. $$f(x)=\frac{x^{2}+2}{x^{2}-16}$$

Use the matrices \(A=\left[\begin{array}{rr}2 & -1 \\ 1 & 3\end{array}\right] \quad\) and \(\quad B=\left[\begin{array}{rr}-1 & 1 \\ 0 & -2\end{array}\right].\) Show that \((A+B)(A-B) \neq A^{2}-B^{2}\)

Determine whether the statement is true or false. Justify your answer. The system $$\left\\{\begin{aligned} x+4 y-5 z &=8 \\ 2 y+z &=5 \\ z &=1 \end{aligned}\right.$$ is in row-echelon form.

Sketch the graph of the function. Identify any asymptotes. $$f(x)=\frac{7}{-x-1}$$

Write the expression as the logarithm of a single quantity. \(\ln x+\ln 6\)

Find the domain of the function and identify any horizontal or vertical asymptotes. $$f(x)=3-\frac{2}{x^{2}}$$

Use the matrices \(A=\left[\begin{array}{rr}2 & -1 \\ 1 & 3\end{array}\right] \quad\) and \(\quad B=\left[\begin{array}{rr}-1 & 1 \\ 0 & -2\end{array}\right].\) Show that \((A+B)^{2}=A^{2}+A B+B A+B^{2}\)

Determine whether the statement is true or false. Justify your answer. If a system of three linear equations is inconsistent, then its graph has no points common to all three equations.

Sketch the graph of the function. Identify any asymptotes. $$f(x)=\frac{4 x}{5 x^{2}+2}$$

Write the expression as the logarithm of a single quantity. \(\ln x-5 \ln (x+3)\)

Find the domain of the function and identify any horizontal or vertical asymptotes. $$f(x)=\frac{x+1}{x^{2}+1}$$

Think About it If \(a, b,\) and \(c\) are real numbers such that \(c \neq 0\) and \(a c=b c,\) then \(a=b .\) However, if \(A, B\) and \(C\) are nonzero matrices such that \(A C=B C\), then \(A\) is not necessarily equal to \(B\). Illustrate this using the following matrices. $$A=\left[\begin{array}{ll} 0 & 1 \\ 0 & 1 \end{array}\right], \quad B=\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right], \quad C=\left[\begin{array}{ll} 2 & 3 \\ 2 & 3 \end{array}\right]$$

You are tutoring a student in algebra. In trying to find a partial fraction decomposition, your student writes the following. $$\begin{aligned} \frac{x^{2}+1}{x(x-1)} &=\frac{A}{x}+\frac{B}{x-1} \\ x^{2}+1 &=A(x-1)+B x \\ x^{2}+1 &=(A+B) x-A \end{aligned}$$ Your student then forms the following system of linear equations. $$\left\\{\begin{aligned} A+B &=0 \\\\-A &=1 \end{aligned}\right.$$ Solve the system and check the partial fraction decomposition it yields. Has your student worked the problem correctly? If not, what went wrong?

Sketch the graph of the function. Identify any asymptotes. $$f(x)=\frac{x^{2}-2 x-3}{x-4}$$

Find the domain of the function and identify any horizontal or vertical asymptotes. $$f(x)=\frac{x-4}{x^{2}+16}$$

Think About It If \(a\) and \(b\) are real numbers such that \(-a b=0,\) then \(a=0\) or \(b=0 .\) However, if \(A\) and \(B\) are matrices such that \(A B=O,\) it is not necessarily true that \(A=O\) or \(B=O .\) Illustrate this using the following matrices. $$A=\left[\begin{array}{ll} 3 & 3 \\ 4 & 4 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right]$$

Sketch the graph of the function. Identify any asymptotes. $$f(x)=\frac{x^{2}-36}{x+1}$$

Write the expression as the logarithm of a single quantity. \(\frac{1}{4} \log _{6} 3+\frac{1}{4} \log _{6} x\)

Let \(i=\sqrt{-1}\) and let \(A=\left[\begin{array}{ll}i & 0 \\ 0 & i\end{array}\right] \quad\) and \(\quad B=\left[\begin{array}{ll}0 & -i \\ i & 0\end{array}\right].\) (a) Find \(A^{2}, A^{3},\) and \(A^{4}\). Identify any similarities with \(i^{2}, i^{3},\) and \(i^{4}.\) (b) Find and identify \(B^{2}.\)

Are the two systems of equations equivalent? Give reasons for your answer. $$\left\\{\begin{aligned} x+3 y-z &=6 \\ 2 x-y+2 z &=1 \\ 3 x+2 y-z &=2 \end{aligned} \quad\left\\{\begin{array}{rr} x+3 y-z= & 6 \\ -7 y+4 z= & 1 \\ -7 y-4 z= & -16 \end{array}\right.\right.$$

Write the expression as the logarithm of a single quantity. \(2 \ln x-\ln (x+2)\)

Let \(A\) and \(B\) be unequal diagonal matrices of the same dimension. (A diagonal matrix is a square matrix in which each entry not on the main diagonal is zero.) Determine the products \(A B\) for several pairs of such matrices. Make a conjecture about a quick rule for such products.

Think About It Let matrices \(A\) and \(B\) be of dimensions \(3 \times 2\) and \(2 \times 2,\) respectively. Answer the following questions and explain your reasoning. (a) Is it possible that \(A=B ?\) (b) Is \(A+B\) defined? (c) Is \(A B\) defined?

Find a system of equations in three variables that has exactly two equations and no solution.

When using Gaussian elimination to solve a system of linear equations, explain how you can recognize that the system has no solution. Give an example that illustrates your answer.