Chapter 6

Find the partial fraction decomposition for each rational expression. $$\frac{5}{3 x(2 x+1)}$$

Graph each inequality. Do not use a calculator. $$x \leq 3$$

Determine whether A and B are inverses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{ll} 5 & 7 \\ 2 & 3 \end{array}\right] ; B=\left[\begin{array}{rr} 3 & -7 \\ -2 & 5 \end{array}\right]$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{rr}-5 & 9 \\\4 & -1\end{array}\right]$$

Find the dimension of each matrix. Identify any square, column, or row matrices. Do not use a calculator. $$\left[\begin{array}{rr} -3 & 6 \\ 7 & -4 \end{array}\right]$$

Verify that the given ordered triple is a solution of the system. Do not use a calculator. \((-3,6,1)\) \begin{aligned}2 x+y-z &=-1 \\\x-y+3 z &=-6 \\\\-4 x+y+z &=19\end{aligned}

Find the partial fraction decomposition for each rational expression. $$\frac{3 x-1}{x(x+1)}$$

Graph each inequality. Do not use a calculator. $$y \leq-2 \quad$$

Determine whether A and B are inverses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{ll} 2 & 3 \\ 1 & 1 \end{array}\right] ; B=\left[\begin{array}{rr} -1 & 3 \\ 1 & -2 \end{array}\right]$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{ll}-1 & 3 \\\\-2 & 9\end{array}\right]$$

Find the dimension of each matrix. Identify any square, column, or row matrices. Do not use a calculator. $$\left[\begin{array}{rrr} 2 & -8 & 6 \\ 1 & 0 & -5 \\ 5 & -2 & 3 \end{array}\right]$$

Use the given row transformation to transform each matrix. Do not use acalculator. $$\left[\begin{array}{rr} -1 & 4 \\ 7 & 0 \end{array}\right] \quad 7 R_{1}$$

Verify that the given ordered triple is a solution of the system. Do not use a calculator. $$\begin{aligned}\left(\frac{1}{2},-\frac{3}{4}, \frac{1}{6}\right)\\\ &2 x+8 y-6 z=-6\\\ &x+y+z=-\frac{1}{12}\\\ &x+3 z=1 \end{aligned}$$}

Find the partial fraction decomposition for each rational expression. $$\frac{4 x+2}{(x+2)(2 x-1)}$$

Graph each inequality. Do not use a calculator. $$x+2 y \leq 6$$

Determine whether A and B are inverses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{rr} -1 & 2 \\ 3 & -5 \end{array}\right] ; B=\left[\begin{array}{rr} -5 & -2 \\ -3 & -1 \end{array}\right]$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{rr}-1 & -2 \\\5 & 3\end{array}\right]$$

Find the dimension of each matrix. Identify any square, column, or row matrices. Do not use a calculator. $$\left[\begin{array}{rrrr} -6 & 8 & 0 & 0 \\ 4 & 1 & 9 & 2 \\ 3 & -5 & 7 & 1 \end{array}\right]$$

Verify that the given ordered triple is a solution of the system. Do not use a calculator. $$\begin{aligned}(-0.2,0.4,0.5)\\\&\begin{aligned} 5 x-y+2 z &=-0.4 \\\x+4 z &=1.8 \\\\-3 y+z &=-0.7\end{aligned}\end{aligned}$$

Find the partial fraction decomposition for each rational expression. $$\frac{x+2}{(x+1)(x-1)}$$

Graph each inequality. Do not use a calculator. $$x-y \geq 2$$

Determine whether A and B are inverses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{ll} 2 & 1 \\ 3 & 2 \end{array}\right] ; B=\left[\begin{array}{rr} 2 & 1 \\ -3 & 2 \end{array}\right]$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{ll}6 & -4 \\\0 & -1\end{array}\right]$$

Find the dimension of each matrix. Identify any square, column, or row matrices. Do not use a calculator. $$\left[\begin{array}{rrrr} -3 & 4 & 2 & 1 \\ 0 & 8 & 6 & 3 \end{array}\right]$$

Verify that the given ordered triple is a solution of the system. Do not use a calculator. \((-1,-2,-3)\) $$x-y+z=-2$$\begin{aligned} x-2 y+z &=0 \\ y-z &=1 \end{aligned}$$

Find the partial fraction decomposition for each rational expression. $$\frac{x}{x^{2}+4 x-5}$$

Determine whether A and B are inverses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{rrr} 0 & 1 & 0 \\ 0 & 0 & -2 \\ 1 & -1 & 0 \end{array}\right] ; B=\left[\begin{array}{rrr} 1 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & -1 & 0 \end{array}\right]$$

Graph each inequality. Do not use a calculator. $$2 x+3 y \geq 4$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{rr}9 & 3 \\\\-3 & -1\end{array}\right]$$

Find the dimension of each matrix. Identify any square, column, or row matrices. Do not use a calculator. $$\left[\begin{array}{l} 2 \\ 4 \end{array}\right]$$

Verify that the given ordered triple is a solution of the system. Do not use a calculator. \((-2,-1,3)\) $$\begin{aligned} x-y+z &=2 \\ 3 x-2 y+z &=-1 \\ x+y &=-3 \end{aligned}$$

Use the given row transformation to transform each matrix. Do not use acalculator. $$\left[\begin{array}{rrr} -3 & 1 & -4 \\ 2 & 1 & 3 \\ 10 & 5 & 2 \end{array}\right]-5 R_{2}+R_{3}$$

Many factors may contribute to population changes in metropolitan areas. The graph shows the populations of the New Orleans, Louisiana, and the Jacksonville, Florida, metropolitan areas over the years \(2004-2009\). If equations of the form \(y=f(t)\) were determined that modeled either of the two graphs, then the variable \(t\) would represent _____ and the variable \(y\) would represent _____.

Find the partial fraction decomposition for each rational expression. $$\frac{5 x-3}{(x+1)(x-3)}$$

Determine whether A and B are inverses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{lll} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{array}\right] ; B=\left[\begin{array}{rrr} 1 & -2 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \end{array}\right]$$

Graph each inequality. Do not use a calculator. $$4 y-3 x<5$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{ll}0 & 2 \\\1 & 5\end{array}\right]$$

Find the dimension of each matrix. Identify any square, column, or row matrices. Do not use a calculator. $$\left[\begin{array}{ll} 4 & 9 \end{array}\right]$$

Verify that the given ordered triple is a solution of the system. Do not use a calculator. \begin{aligned} &\text\\\\\left(\frac{1}{2}, \frac{1}{2},-2\right)\\\ &3 x+y+z=0\\\ &4 x+2 y+z=1\\\ &2 x-2 y-z=2 \end{aligned}

Find the partial fraction decomposition for each rational expression. $$\frac{2 x}{(x+1)(x+2)^{2}}$$

Determine whether A and B are inverses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{rrr} -1 & -1 & -1 \\ 4 & 5 & 0 \\ 0 & 1 & -3 \end{array}\right] ; B=\left[\begin{array}{rrr} 15 & 4 & -5 \\ -12 & -3 & 4 \\ -4 & -1 & 1 \end{array}\right]$$

Graph each inequality. Do not use a calculator. $$3 x-5 y>6$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{rr}3 & 4 \\\5 & -2\end{array}\right]$$

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 33-36, let \(t=\frac{1}{x}, u=\frac{1}{y},\) and \(v=\frac{1}{z} .\) Solve for \(t, u,\) and \(v,\) and then find \(x, y, \text { and } z .)\)\begin{aligned}x+y+z &=2 \\\2 x+y-z &=5 \\\x-y+z &=-2\end{aligned}

Find the dimension of each matrix. Identify any square, column, or row matrices. Do not use a calculator. $$[-9]$$

Write the augmented matrix for each system. Do not solve the system. $$\begin{array}{l} 2 x+3 y=11 \\ x+2 y=8 \end{array}$$

Find the partial fraction decomposition for each rational expression. $$\frac{2}{x^{2}(x+3)}$$

Determine whether A and B are inverses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{lll} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{array}\right] ; B=\left[\begin{array}{rrr} 7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right]$$

Graph each inequality. Do not use a calculator. $$x<3+2 y$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{rr}-9 & 7 \\\2 & 6\end{array}\right]$$