Chapter 7

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{v^{2}-5 v+4}{v^{2}-6 v+8}}{\frac{v^{2}+2 v-3}{v^{2}+v-6}} $$

For exercises 13-24, rewrite each expression as an equivalent expression with the given denominator. $$ \frac{2 p}{p^{2}-36} ;(p+6)(p-6)(p+1) $$

For exercises \(5-48\), simplify. $$ \frac{w^{2}}{w+8}-\frac{64}{w+8} $$

For exercises 1-66, simplify. $$ \frac{5}{5 x^{2}+10 x} $$

If the force acting on an object is constant, the relationship of the mass of the object, \(x\), and the acceleration of the object, \(y\), is an inverse variation. When the mass is \(1000 \mathrm{~kg}\), the acceleration is \(\frac{4 \mathrm{~m}}{1 \mathrm{~s}^{2}}\). a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the acceleration when the mass is \(1500 \mathrm{~kg}\). Round to the nearest tenth.

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{p^{2}-11 p+30}{p^{2}-2 p-24}}{\frac{p^{2}-4 p-5}{p^{2}+5 p+4}} $$

For exercises \(5-48\), simplify. $$ \frac{k^{2}}{k+7}-\frac{49}{k+7} $$

For exercises 7-32, simplify. $$ \frac{c^{2}+18 c+81}{c^{2}-4 c+4} \cdot \frac{c^{2}-5 c+6}{c^{2}+6 c-27} $$

The relationship of the number of tickets sold, \(x\), and the total ticket receipts for an outdoor concert, \(y\), is a direct variation. When 11,000 tickets are sold, the total ticket receipts are \(\$ 495,000\). a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the number of tickets sold when the total ticket receipts are \(\$ 562,500\). d. Find the total ticket receipts from the sale of 7575 tickets. e. What does \(k\) represent in this equation?

For exercises 13-24, rewrite each expression as an equivalent expression with the given denominator. $$ \frac{3 c}{c^{2}-13 c+40} ;(c-8)(c-2)(c-5) $$

For exercises 7-32, simplify. $$ \frac{2 r^{2}-4 r-6}{r^{2}+5 r-24} \cdot \frac{r+8}{2 r} $$

For exercises 1-66, simplify. $$ \frac{3 x-6}{4 x-8} $$

The relationship of the taxable value of a property, \(x\), and the annual property tax, \(y\), is a direct variation. When the taxable value of a property is \(\$ 250,000\), the annual property tax bill is \(\$ 5375\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this relationship. c. Find the taxable value of a property with an annual property tax bill of \(\$ 8062.50\). d. Find the tax owed for a property with an assessed value of \(\$ 185,000\). Round to the nearest whole number. e. What does \(k\) represent in this equation?

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{1}{z^{2}+9 z+14}}{z^{2}-49} $$

For exercises \(5-48\), simplify. $$ \frac{k}{k^{2}-49}-\frac{7}{k^{2}-49} $$

For exercises 7-32, simplify. $$ \frac{3 d^{2}+9 d-12}{d^{2}+10 d+24} \cdot \frac{d+6}{3 d} $$

For exercises 1-66, simplify. $$ \frac{5 x-15}{9 x-27} $$

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{2}+\frac{1}{3}}{\frac{1}{3}+\frac{1}{5}} $$

For exercises \(5-48\), simplify. $$ \frac{w}{w^{2}+64}-\frac{8}{w^{2}+64} $$

For exercises 7-32, simplify. $$ \frac{m^{2}-2 m-80}{m^{2}-m-90} \cdot \frac{m^{2}+6 m-27}{m^{2}+5 m-24} $$

The relationship of the time a tour guide works, \(x\), and the cost to hire the tour guide, \(y\), is a direct variation. When a tour guide works for \(15 \mathrm{hr}\), the cost is \(\$ 1125\). a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the cost to hire a tour guide for \(8 \mathrm{hr}\). d. What does \(k\) represent in this equation?

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{3}+\frac{1}{2}}{\frac{1}{2}+\frac{1}{7}} $$

For exercises \(5-48\), simplify. $$ \frac{k}{k^{2}+49}-\frac{7}{k^{2}+49} $$

For exercises 7-32, simplify. $$ \frac{a^{2}+7 a-44}{a^{2}+9 a-22} \cdot \frac{a^{2}-9 a+14}{a^{2}-11 a+28} $$

The relationship of the distance driven, \(x\), and the cost of gasoline, \(y\), is a direct variation. For a trip of \(400 \mathrm{mi}\), the cost is \(\$ 60\). a. Find the constant of proportionality. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the cost of gasoline to drive \(225 \mathrm{mi}\). d. What does \(k\) represent in this equation?

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{2}}{\frac{1}{3}+\frac{1}{5}} $$

For exercises 27-34, evaluate. $$ \frac{1}{12}+\frac{5}{12} $$

For exercises \(5-48\), simplify. $$ \frac{x^{2}}{x-9}-\frac{7 x+18}{x-9} $$

For exercises 7-32, simplify. $$ \frac{2 x^{2}-5 x-3}{x^{2}-12 x+27} \cdot \frac{x^{2}-15 x+54}{2 x^{2}+13 x+6} $$

For exercises 1-66, simplify. $$ \frac{6 x+6}{3 x-3} $$

The relationship of the distance driven, \(x\), and the cost of gasoline, \(y\), is a direct variation. For a trip of \(250 \mathrm{mi}\), the cost is \(\$ 90\). a. Find the constant of proportionality. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the cost of gasoline to drive \(225 \mathrm{mi}\). d. What does \(k\) represent in this equation?

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{3}}{\frac{1}{2}+\frac{1}{7}} $$

For exercises 27-34, evaluate. $$ \frac{1}{14}+\frac{5}{14} $$

For exercises \(5-48\), simplify. $$ \frac{x^{2}}{x-7}-\frac{3 x+28}{x-7} $$

For exercises 7-32, simplify. $$ \frac{3 x^{2}+14 x+8}{x^{2}-5 x-36} \cdot \frac{x^{2}-4 x-45}{3 x^{2}-13 x-10} $$

For exercises 1-66, simplify. $$ \frac{8 x+8}{4 x-4} $$

The relationship of the distance driven, \(x\), and the cost of gasoline, \(y\), is a direct variation. For a trip of \(250 \mathrm{mi}\), the cost is \(\$ 90\). a. Find the constant of proportionality. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the cost of gasoline to drive \(225 \mathrm{mi}\). d. What does \(k\) represent in this equation?

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{2}{5}-\frac{1}{3}}{\frac{1}{5}+\frac{1}{3}} $$

For exercises \(5-48\), simplify. $$ \frac{z^{2}}{z+3}-\frac{5 z+24}{z+3} $$

For exercises 7-32, simplify. $$ \frac{12 r^{5}+60 r^{4}}{r^{4}-r^{3}} \cdot \frac{r^{2}-1}{27 r+135} $$

When a car travels a fixed distance, the relationship between the speed of the car, \(x\), and the time it travels, \(y\), is an inverse variation. When the speed is \(\frac{48 \mathrm{mi}}{1 \mathrm{hr}}\), the time is \(0.75 \mathrm{hr}\). a. Find the constant of proportionality. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the time in hours to travel this distance at a speed of \(\frac{80 \mathrm{mi}}{1 \mathrm{hr}}\). d. Change the time in part \(\mathrm{c}\) to minutes.

For exercises 11-30, (a) solve. (b) check. $$ \frac{4}{x-7}=\frac{24}{2 x-6} $$

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{2}{3}-\frac{1}{2}}{\frac{1}{3}+\frac{1}{6}} $$

For exercises 7-32, simplify. $$ \frac{28 x^{6}+42 x^{5}}{x^{3}-x^{2}} \cdot \frac{x^{2}-1}{42 x+63} $$

In radiography, a grid reduces the effect of X-ray scattering. The relationship of the interspace distance on the grid, \(x\), and the grid ratio, \(y\), is an inverse variation. When the interspace distance on a grid is 300 micrometers, the grid ratio is 8 . Write an equation that represents this variation. Include the units.

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{4}{9}+\frac{3}{2}}{\frac{5}{9}-\frac{1}{6}} $$

For exercises 27-34, evaluate. $$ \frac{5}{9}+\frac{7}{24} $$

For exercises \(5-48\), simplify. $$ \frac{c^{2}}{2 c-16}-\frac{10 c-16}{2 c-16} $$

For exercises 7-32, simplify. $$ \frac{w^{2}-3 w+5}{w^{2}-4} \cdot \frac{w^{2}+10 w+16}{w^{2}-1} $$

When the radiation is constant, the relationship of the current in an X-ray tube, \(x\), and the exposure time, \(y\), is an inverse variation. When the current is 600 milliamp, the exposure time is \(0.2 \mathrm{~s}\). Write an equation that represents this variation. Include the units.