Chapter 7

ⓐ $x=0$ ⓑ $n=-\frac{1}{3}$
ⓒ $a=\mathrm{-1},a=\mathrm{-3}$

ⓐ $q=0$ ⓑ $y=-\frac{2}{3}$
ⓒ $m=2,m=\mathrm{-3}$

$\frac{x+1}{x-1},$ $x\ne 2,$ $x\ne 1$

$\frac{x-5}{x-1},$ $x\ne \text{−}2,$ $x\ne 1$

$\frac{2\left(x-3y\right)}{3\left(x+3y\right)}$

$\frac{5\left(x-y\right)}{2\left(x+5y\right)}$

$-\frac{x+1}{x+5}$

$-\frac{x+2}{x+1}$

$\frac{x-2}{2\left(x+3\right)}$

$\frac{3(x-6)}{x+5}$

$\frac{x-4}{x-5}$

$-\frac{(b+2)(b-1)}{(1+b)(b+4)}$

$\frac{2\left(x^2+2x+4\right)}{\left(x+2\right)\left(x^2-2x+4\right)}$

$\frac{2z}{z-1}$

$\frac{x+2}{4}$

$\frac{2}{y+5}$

$\frac{2(m+1)(m+2)}{3(m+4)(m-3)}$

$\frac{(n+5)(n+9)}{2(n+6)(2n+3)}$

The domain of $R\left(x\right)$ is all real numbers where $x\ne 5$ and $x\ne \text{−}1.$

The domain of $R\left(x\right)$ is all real numbers where $x\ne 4$ and $x\ne \text{−}2.$

$R\left(x\right)=2$

$R\left(x\right)=\frac{1}{3}$

$R\left(x\right)=\frac{x-2}{4\left(x-8\right)}$

$R\left(x\right)=\frac{x(x-2)}{x-1}$

$x+2$

$x+3$

$\frac{x-11}{x-2}$

$\frac{x-3}{x+9}$

$\frac{y+3}{y+2}$

$\frac{3n-2}{n-1}$

ⓐ $\left(x-4\right)\left(x+3\right)\left(x+4\right)$
ⓑ $\frac{2x+8}{\left(x-4\right)\left(x+3\right)\left(x+4\right)}$,
$\frac{x+3}{\left(x-4\right)\left(x+3\right)\left(x+4\right)}$

ⓐ $\left(x+2\right)\left(x-5\right)\left(x+1\right)$
ⓑ $\frac{3x^2+3x}{\left(x+2\right)\left(x-5\right)\left(x+1\right)}$,
$\frac{5x-25}{\left(x+2\right)\left(x-5\right)\left(x+1\right)}$

$\frac{7x-4}{(x-2)(x+3)}$

$\frac{7m+25}{(m+3)(m+4)}$

$\frac{5m^2-9m+2}{(m+1)(m-2)(m+2)}$

$\frac{2n^2+12n-30}{(n+2)(n-5)(n+3)}$

$\frac{1}{x-2}$

$\frac{\mathrm{-3}}{z-3}$

$\frac{5x+1}{\left(x-6\right)\left(x+1\right)}$

$\frac{y+3}{y+4}$

$\frac{1}{\left(b+1\right)\left(b-1\right)}$

$\frac{1}{\left(x+2\right)\left(x+1\right)}$

$\frac{v+3}{v+1}$

$\frac{3w}{w+7}$

$\frac{x-7}{x-4}$

$\frac{x^2-3x+18}{\left(x+3\right)\left(x-3\right)}$

$\frac{2}{3\left(x-1\right)}$

$\frac{1}{2\left(x-3\right)}$

$\frac{14}{11}$

$\frac{10}{23}$

$\frac{y+x}{y-x}$

$\frac{ab}{b-a}$

$\frac{b\left(b+2\right)\left(b-5\right)}{3b-5}$

$\frac{3}{c+3}$

$\frac{7}{3}$

$\frac{10}{3}$

$\frac{b+a}{a^2+b^2}$

$\frac{y-x}{xy}$

$\frac{3\left(x-2\right)}{5x+7}$

$\frac{x+21}{6x-43}$

$\frac{3}{5x+22}$

$\frac{2\left(2y^2+13y+5\right)}{3y}$

$\frac{x}{x+4}$

$\frac{x\left(x+1\right)}{3\left(x-1\right)}$

$y=-\frac{7}{15}$

$x=\frac{13}{15}$

$x=\mathrm{-3},x=5$

$y=\mathrm{-2},y=6$

$x=\frac{2}{3}$

$y=2$

There is no solution.

There is no solution.

$x=3$

$y=7$

There is no solution.

There is no solution.

There is no solution.

There is no solution.

ⓐ The domain is all real numbers except $x\ne 3$ and $x\ne 4.$ ⓑ $x=2,x=\frac{14}{3}$
ⓒ $\left(2,3\right),\left(\frac{14}{3},3\right)$

ⓐ The domain is all real numbers except $x\ne 1$ and $x\ne 5.$ ⓑ $x=\frac{21}{4}$ ⓒ $\left(\frac{21}{4},4\right)$

$y=mx-4m+5$

$y=mx+5m+1$

$a=\frac{b}{cb-1}$

$y=\frac{3x}{x+6}$

$y=33$

$z=14$

The pediatrician will prescribe 12 ml of acetaminophen to Emilia.

The pediatrician will prescribe 180 mg of fever reducer to Isabella.

The distance is 150 miles.

The distance is 350 miles.

The telephone pole is 40 feet tall.

The pine tree is 60 feet tall.

Link’s biking speed is 15 mph.

The speed of Danica’s boat is 17 mph.

Dennis’s uphill speed was 10 mph and his downhill speed was 5 mph.

Joon’s rate on the country roads is 50 mph.

Kayla’s biking speed was 15 mph.

Victoria jogged 6 mph on the flat trail.

When the two gardeners work together it takes 2 hours and 24 minutes.

When Daria and her mother work together it takes 2 hours and 6 minutes.

Kristina can paint the room in 12 hours.

It will take Jordan 6 hours.

ⓐ $c=4.8t$ ⓑ He would burn 432 calories.

ⓐ $d=50t$ ⓑ It would travel 250 miles.

ⓐ $h=\frac{130}{t}$ ⓑ $1\frac{2}{3}$ hours

ⓐ $x=\frac{3500}{p}$ ⓑ 500 units

$(\text{−}\infty ,\mathrm{-4})\cup [2,\infty )$

$(\text{−}\infty ,\mathrm{-2}]\cup (4,\infty )$

$\left(-\frac{3}{2},3\right)$

$\left(\mathrm{-8},4\right)$

$\left(\text{−}\infty ,\mathrm{-4}\right)\cup \left(2,\infty \right)$

$\left(\text{−}\infty ,\mathrm{-4}\right)\cup \left(3,\infty \right)$

$(2,4)$

$(3,6)$

$(\mathrm{-4},2]$

$[\mathrm{-1},4)$

ⓐ $c(x)=\frac{20x+6000}{x}$
ⓑ More than 150 items must be produced to keep the average cost below $60 per item.

ⓐ $c(x)=\frac{5x+900}{x}$ ⓑ More than 60 items must be produced to keep the average cost below $20 per item.

ⓐ $z=0$ ⓑ $p=\frac{5}{6}$
ⓒ $n=\mathrm{-4},n=2$

ⓐ $y=0$, ⓑ $x=-\frac{1}{2}$, ⓒ $u=\mathrm{-4},u=7$

$-\frac{4}{5}$

$\frac{2m^2}{3n}$

$\frac{8}{3}$

$\frac{x+5}{x-1}$

$\frac{a+2}{a+8}$

$\frac{p^2+4}{p-2}$

$\frac{4b(b-4)}{(b+5)(b-8)}$

$\frac{3(m+5n)}{4(m-5n)}$

$\mathrm{-1}$

$-\frac{5}{y+4}$

$\frac{w^2-6w+36}{w-6}$

$-\frac{z-5}{4+z}$

$\frac{3}{10}$

$\frac{x^3}{8y}$

$\frac{p(p-4)}{2\left(p-9\right)}$

$\frac{y-5}{3(y+5)}$

$-\frac{4(b+9)}{3(b+7)}$

$\frac{c+5}{3c+1}$

$\frac{(m-3)(m-2)}{(m+4)(m+3)}$

$-\frac{1}{v+5}$

$\frac{3s}{s+4}$

$\frac{4(p^2-pq+q^2)}{(p-q)(p^2+pq+q^2)}$

$\frac{x-2}{8x(x+5)}$

$\frac{2a-7}{5}$

$3\left(3c-5\right)$

$\frac{4(m+8)(m+7)}{3(m-4)(m+2)}$

$\frac{(4p+1)(p-4)}{3p(p+9)(p-1)}$

$x\ne 5$ and $x\ne \text{−}5$

$x\ne 2$ and $x\ne \text{−}3$

$R\left(x\right)=2$

$R\left(x\right)=\frac{x+5}{2x(x+2)}$

$R\left(x\right)=\frac{3x(x+7)}{x-7}$

$R\left(x\right)=\frac{x(x-5)}{x-6}$

Answers will vary.

Answers will vary.

$\frac{3}{5}$

$\frac{3c+5}{4c-5}$

$r+8$

$\frac{2w}{w-4}$

$3a+7$

$\frac{m-2}{2}$

$\frac{p+3}{p+5}$

$\frac{r+9}{r+7}$

$4$

$x+2$

$\frac{z+4}{z-5}$

$\frac{4b-3}{b-7}$

ⓐ $\left(x+2\right)\left(x-4\right)\left(x+3\right)$
ⓑ $\frac{5x+15}{\left(x+2\right)\left(x-4\right)\left(x+3\right)}$,
$\frac{2x^2+4x}{\left(x+2\right)\left(x-4\right)\left(x+3\right)}$

ⓐ $\left(z-2\right)\left(z+4\right)\left(z-4\right)$
ⓑ $\frac{9z-36}{\left(z-2\right)\left(z+4\right)\left(z-4\right)}$,
$\frac{4z^2-8z}{\left(z-2\right)\left(z+4\right)\left(z-4\right)}$

ⓐ $\left(b+3\right)\left(b+3\right)\left(b-5\right)$
ⓑ $\frac{4b-20}{\left(b+3\right)\left(b+3\right)\left(b-5\right)}$,
$\frac{2b^2+6b}{\left(b+3\right)\left(b+3\right)\left(b-5\right)}$

ⓐ $\left(d+5\right)\left(3d-1\right)\left(d-6\right)$
ⓑ $\frac{2d-12}{\left(d+5\right)\left(3d-1\right)\left(d-6\right)}$,
$\frac{5d^2+25d}{\left(d+5\right)\left(3d-1\right)\left(d-6\right)}$

$\frac{21y+8x}{30x^2{y}^2}$

$\frac{5r-7}{(r+4)(r-5)}$

$\frac{11w+1}{(3w-2)(w+1)}$

$\frac{2y^2+y+9}{(y+3)(y-1)}$

$\frac{b(5b+10+2a^2)}{a^2(b-2)(b+2)}$

$\text{−}\frac{m}{m+4}$

$\frac{3(r^2+6r+18)}{(r+1)(r+6)(r+3)}$

$\frac{2(7t-6)}{(t-6)(t+6)}$

$\frac{4a^2+25a-6}{(a+3)(a+6)}$

$\frac{\mathrm{-6}}{m-6}$

$\frac{p+2}{p+3}$

$\frac{3}{r-2}$

$\frac{4(8x+1)}{10x-1}$

$\frac{x-5}{\left(x-4\right)\left(x+1\right)\left(x-1\right)}$

$\frac{1}{\left(x-1\right)\left(x+1\right)}$

$\frac{5a^2+7a-36}{a\left(a-2\right)}$

$\frac{c-5}{c+2}$

$\frac{3(d+1)}{d+2}$

ⓐ $R\left(x\right)=-\frac{\left(x+8\right)\left(x+1\right)}{\left(x-2\right)\left(x+3\right)}$ ⓑ $R\left(x\right)=\frac{x+1}{x+3}$

ⓐ $\frac{3\left(3x+8\right)}{\left(x-8\right)\left(x+8\right)}$
ⓑ $R\left(x\right)=\frac{3}{x+8}$

Answers will vary.

ⓐ Answers will vary.
ⓑ Answers will vary.
ⓒ Answers will vary.
ⓓ $\frac{x+y}{xy}$

$\frac{a-4}{2a}$

$\frac{1}{2(c-2)}$