13.E: Review Exercises 2

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Exercise $\PageIndex{1}$

Evaluate

Given $f ( x ) = 2 x ^ { 2 } - x + 6$, find $f ( - 3 ) , f ( 0 )$, and $f ( 10 )$.
Given $g ( x ) = - x ^ { 2 } + 4 x - 1$, find $g ( - 1 ) , g ( 0 )$, and $g(3)$.
Given $h ( t ) = - t ^ { 3 } - 2 t ^ { 2 } + 3$, find $h ( - 3 ) , h ( 0 )$, and $h(2)$.
Given $p ( x ) = x ^ { 4 } - 2 x ^ { 2 } + x$, find $p ( - 1 ) , p ( 0 )$, and $p(2)$.
The following graph gives the height $h (t)$ in feet of a projectile over time $t$ in seconds

(a) Use the graph to determine the height of the projectile at $2.5$ seconds.

(b) At what time does the projectile reach its maximum height?

6. Given the graph of the function $f$, find $f ( - 9 ) , f ( - 3 )$, and $f(12)$.

7. From the ground, a bullet is fired straight up into the air at $340$ meters per second. Ignoring the effects of air friction, write a function that models the height of the bullet and use it to calculate the bullet’s height after one-quarter of a second. (Round off to the nearest meter.)

8. An object is tossed into the air at an initial speed of $30$ feet per second from a rooftop $10$ feet high. Write a function that models the height of the object and use it to calculate the height of the object after $1$ second.

Answer

1. $f ( - 3 ) = 27 ; f ( 0 ) = 6 ; f ( 10 ) = 196$

3. $h ( - 3 ) = 12 ; h ( 0 ) = 3 ; h ( 2 ) = - 13$

5. (a) $60$ feet; (b) $2$ seconds; (c) $4$ seconds

7. $h ( t ) = - 4.9 t ^ { 2 } + 340 t$; at $0.25$ second, the bullet's height is about $85$ meters.

Exercise $\PageIndex{2}$

Perform the operations.

Given $f ( x ) = 5 x ^ { 2 } - 3 x + 1$ and $g ( x ) = 2 x ^ { 2 } - x - 1$, find $( f + g ) ( x )$.
Given $f ( x ) = x ^ { 2 } + 3 x - 8$ and $g ( x ) = x ^ { 2 } - 5 x - 7$, find $( f - g ) ( x )$.
Given $f ( x ) = 3 x ^ { 2 } - x + 2$ and $g ( x ) = 2 x - 3$, find $( f \cdot g ) ( x )$.
Given $f ( x ) = 27 x ^ { 5 } - 15 x ^ { 3 } - 3 x ^ { 2 }$ and $g ( x ) = 3 x ^ { 2 }$, find $( f / g ) ( x )$.
Given $g ( x ) = x ^ { 2 } - x + 1$, find $g ( - 3 u )$.
GIven $g ( x ) = x ^ { 3 } - 1$, find $g ( x - 1 )$.

Answer

1. $( f + g ) ( x ) = 7 x ^ { 2 } - 4 x$

3. $( f \cdot g ) ( x ) = 6 x ^ { 3 } - 11 x ^ { 2 } + 7 x - 6$

5. $g ( - 3 u ) = 9 u ^ { 2 } + 3 u + 1$

Exercise $\PageIndex{3}$

Given $f ( x ) = 16 x ^ { 3 } - 12 x ^ { 2 } + 4 x , g ( x ) = x ^ { 2 } - x + 1$, and $h ( x ) = 4 x$, find the following:

$( g \cdot h ) ( x )$
$( f - g ) ( x )$
$( g + f ) ( x )$
$( f / h ) ( x )$
$( f \cdot h ) ( - 1 )$
$( g + h ) ( - 3 )$
$( g - f ) ( 2 )$
$( f / h ) \left( \frac { 3 } { 2 } \right)$

Answer

1. $( g \cdot h ) ( x ) = 4 x ^ { 3 } - 4 x ^ { 2 } + 4 x$

3. $( g + f ) ( x ) = 16 x ^ { 3 } - 11 x ^ { 2 } + 3 x + 1$

5. $( f \cdot h ) ( - 1 ) = 128$

7. $( g - f ) ( 2 ) = - 85$

Exercise $\PageIndex{4}$

Factor our the greatest common factor (GCF).

$2 x ^ { 4 } - 12 x ^ { 3 } - 2 x ^ { 2 }$
$18 a ^ { 3 } b - 3 a ^ { 2 } b ^ { 2 } + 3 a b ^ { 3 }$
$x ^ { 4 } y ^ { 3 } - 3 x ^ { 3 } y + x ^ { 2 } y$
$x ^ { 3 n } - x ^ { 2 n } - x ^ { n }$

Answer

1. $2 x ^ { 2 } \left( x ^ { 2 } - 6 x - 1 \right)$

3. $x ^ { 2 } y \left( x ^ { 2 } y ^ { 2 } - 3 x + 1 \right)$

Exercise $\PageIndex{5}$

Factor by grouping.

$2 x ^ { 3 } - x ^ { 2 } + 2 x - 1$
$3x ^ { 3 } - x ^ { 2 } - 6 x + 2$
$x ^ { 3 } - 5 x ^ { 2 } y + x y ^ { 2 } - 5 y ^ { 3 }$
$a ^ { 2 } b - a + a b ^ { 3 } - b ^ { 2 }$
$2 x ^ { 4 } - 4 x y ^ { 3 } + 2 x ^ { 2 } y ^ { 2 } - 4 x ^ { 3 } y$
$x ^ { 4 } y ^ { 2 } - x y ^ { 5 } + x ^ { 3 } y ^ { 4 } - x ^ { 2 } y ^ { 3 }$

Answer

1. $\left( x ^ { 2 } + 1 \right) ( 2 x - 1 )$

3. $\left( x ^ { 2 } + y ^ { 2 } \right) ( x - 5 y )$

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

Exercise $\PageIndex{6}$

Factor the special binomials.

$64 x ^ { 2 } - 1$
$9 - 100 y ^ { 2 }$
$x ^ { 2 } - 36 y ^ { 2 }$
$4 - ( 2 x - 1 ) ^ { 2 }$
$a ^ { 3 } b ^ { 3 } + 125$
$64 x ^ { 3 } - y ^ { 3 }$
$81 x ^ { 4 } - y ^ { 4 }$
$x ^ { 8 } - 1$
$x ^ { 6 } - 64 y ^ { 6 }$
$1 - a ^ { 6 } b ^ { 6 }$

Answer

1. $( 8 x + 1 ) ( 8 x - 1 )$

3. $( x + 6 y ) ( x - 6 y )$

5. $( a b + 5 ) \left( a ^ { 2 } b ^ { 2 } - 5 a b + 25 \right)$

7. $\left( 9 x ^ { 2 } + y ^ { 2 } \right) ( 3 x + y ) ( 3 x - y )$

9. $( x + 2 y ) \left( x ^ { 2 } - 2 x y + 4 y ^ { 2 } \right) ( x - 2 y ) \left( x ^ { 2 } + 2 x y + 4 y ^ { 2 } \right)$

Exercise $\PageIndex{7}$

Factor.

$x ^ { 2 } - 8 x - 48$
$x ^ { 2 } - 15 x + 54$
$x ^ { 2 } - 4 x - 6$
$x ^ { 2 } - 12 x y + 36 y ^ { 2 }$
$x ^ { 2 } + 20 x y + 75 y ^ { 2 }$
$- x ^ { 2 } + 5 x + 150$
$- 2 y ^ { 2 } + 20 y + 48$
$28 x ^ { 2 } + 20 x + 3$
$150 x ^ { 2 } - 100 x + 6$
$24 a ^ { 2 } - 38 a b + 3 b ^ { 2 }$
$27 u ^ { 2 } - 3 u v - 4 v ^ { 2 }$
$16 x ^ { 2 } y ^ { 2 } - 78 x y + 27$
$16 m ^ { 2 } + 72 m n + 81 n ^ { 2 }$
$4 x ^ { 2 } - 5 x + 20$
$25 x ^ { 4 } - 35 x ^ { 2 } + 6$
$2 x ^ { 4 } + 7 x ^ { 2 } + 3$
$x ^ { 6 } + 3 x ^ { 3 } y ^ { 3 } - 10 y ^ { 6 }$
$a ^ { 6 } - 8 a ^ { 3 } b ^ { 3 } + 15 b ^ { 6 }$
$x ^ { 2 n } - 2 x ^ { n } + 1$
$6 x ^ { 2 n } - x ^ { n } - 2$

Answer

1. $( x - 12 ) ( x + 4 )$

3. Prime

5. $( x + 5 y ) ( x + 15 y )$

7. $- 2 ( y - 12 ) ( y + 2 )$

9. $2 ( 15 x - 1 ) ( 5 x - 3 )$

11. $( 3 u + v ) ( 9 u - 4 v )$

13. $( 4 m + 9 n ) ^ { 2 }$

15. $\left( 5 x ^ { 2 } - 6 \right) \left( 5 x ^ { 2 } - 1 \right)$

17. $\left( x ^ { 3 } + 5 y ^ { 3 } \right) \left( x ^ { 3 } - 2 y ^ { 3 } \right)$

19. $\left( x ^ { n } - 1 \right) ^ { 2 }$

Exercise $\PageIndex{8}$

Factor completely.

$45 x ^ { 3 } - 20 x$
$12 x ^ { 4 } - 70 x ^ { 3 } + 50 x ^ { 2 }$
$- 20 x ^ { 2 } + 32 x - 3$
$- x ^ { 3 } y + 9 x y ^ { 3 }$
$24 a ^ { 4 } b ^ { 2 } + 3 a b ^ { 5 }$
$64 a ^ { 6 } b ^ { 6 } - 1$
$64 x ^ { 2 } + 1$
$x ^ { 3 } + x ^ { 2 } y - x y ^ { 2 } - y ^ { 3 }$

Answer

1. $5 x ( 3 x + 2 ) ( 3 x - 2 )$

3. $- ( 10 x - 1 ) ( 2 x - 3 )$

5. $3 a b ^ { 2 } ( 2 a + b ) \left( 4 a ^ { 2 } - 2 a b + b ^ { 2 } \right)$

7. Prime

Exercise $\PageIndex{9}$

Solve by factoring.

$9 x ^ { 2 } + 8 x = 0$
$x ^ { 2 } - 1 = 0$
$x ^ { 2 } - 12 x + 20 = 0$
$x ^ { 2 } - 2 x - 48 = 0$
$( 2 x + 1 ) ( x - 2 ) = 3$
$2 - ( x - 4 ) ^ { 2 } = - 7$
$( x - 6 ) ( x + 3 ) = - 18$
$( x + 5 ) ( 2 x - 1 ) = 3 ( 2 x - 1 )$
$\frac { 1 } { 2 } x ^ { 2 } + \frac { 2 } { 3 } x - \frac { 1 } { 8 } = 0$
$\frac { 1 } { 4 } x ^ { 2 } - \frac { 19 } { 12 } x + \frac { 1 } { 2 } = 0$
$x ^ { 3 } - 2 x ^ { 2 } - 24 x = 0$
$x ^ { 4 } - 5 x ^ { 2 } + 4 = 0$

Answer

1. $-\frac{8}{9} , 0$

3. $2,10$

5. $-1, \frac{5}{2}$

7. $0,3$

9. $- \frac { 3 } { 2 } , \frac { 1 } { 6 }$

11. $- 4,0,6$

Exercise $\PageIndex{10}$

Find the roots of the given functions.

$f ( x ) = 12 x ^ { 2 } - 8 x$
$g ( x ) = 2 x ^ { 3 } - 18 x$
$h ( t ) = - 16 t ^ { 2 } + 64$
$p ( x ) = 5 x ^ { 2 } - 21 x + 4$

7. The height in feet of an object dropped from the top of a $16$-foot ladder is given by $h ( t ) = - 16 t ^ { 2 } + 16$, where $t$ represents the time in seconds after the object has been dropped. How long will it take to hit the ground?

8. The length of a rectangle is $2$ centimeters less than twice its width. If the area of the rectangle is $112$ square centimeters, find its dimensions.

9. A triangle whose base is equal in measure to its height has an area of $72$ square inches. Find the length of the base.

10. A box can be made by cutting out the corners and folding up the edges of a sheet of cardboard. A template for a rectangular cardboard box of height $2$ inches is given.

What are the dimensions of a cardboard sheet that will make a rectangular box with volume $240$ cubic inches?

Answer

1. $0 , \frac { 2 } { 3 }$

3. $\pm 2$

5. $- 9,0,6$

7. $1$ second

9. $12$ inches

Exercise $\PageIndex{11}$

Solve or factor.

$x ^ { 2 } - 25$
$x ^ { 2 } - 121 = 0$
$16 x ^ { 2 } - 22 x - 3 = 0$
$3 x ^ { 2 } - 14 x - 5$
$x ^ { 3 } - x ^ { 2 } - 2 x - 2$
$3 x ^ { 2 } = - 15 x$

Answer

1. Factor; $( x + 5 ) ( x - 5 )$

3. Solve; $- \frac { 1 } { 8 } , \frac { 3 } { 2 }$

5. Factor; $( x - 1 ) \left( x ^ { 2 } - 2 \right)$

Exercise $\PageIndex{12}$

Find a polynomial equation with integer coefficients, given the solutions.

$5, -2$
$\frac { 2 } { 3 } , - \frac { 1 } { 2 }$
$\pm \frac { 4 } { 5 }$
$\pm 10$
$-4,0,3$
$-8$ double root

Answer

1. $x ^ { 2 } - 3 x - 10 = 0$

3. $25 x ^ { 2 } - 16 = 0$

5. $x ^ { 3 } + x ^ { 2 } - 12 x = 0$

Exercise $\PageIndex{13}$

State the restrictions and simplify.

$\frac { 108 x ^ { 3 } } { 12 x ^ { 2 } }$
$\frac { 56 x ^ { 2 } ( x - 2 ) ^ { 2 } } { 8 x ( x - 2 ) ^ { 3 } }$
$\frac { 64 - x ^ { 2 } } { 2 x ^ { 2 } - 15 x - 8 }$
$\frac { 3 x ^ { 2 } + 28 x + 9 } { 81 - x ^ { 2 } }$
$\frac { x ^ { 2 } - 25 } { 5 x ^ { 2 } } \cdot \frac { 10 x ^ { 2 } - 15 x } { 2 x ^ { 2 } + 7 x - 15 }$
$\frac { 7 x ^ { 2 } - 41 x - 6 } { ( x - 7 ) ^ { 2 } } \cdot \frac { 49 - x ^ { 2 } } { x ^ { 2 } + x - 42 }$
$\frac { 28 x ^ { 2 } ( 2 x - 3 ) } { 4 x ^ { 2 } - 9 } \div \frac { 7 x } { 4 x ^ { 2 } - 12 x + 9 }$
$\frac { x ^ { 2 } - 10 x + 24 } { x ^ { 2 } - 8 x + 16 } \div \frac { 2 x ^ { 2 } - 13 x + 6 } { x ^ { 2 } + 2 x - 24 }$

Answer

1. $9 x ; x \neq 0$

3. $- \frac { x + 8 } { 2 x + 1 } ; x \neq - \frac { 1 } { 2 } , 8$

5. $\frac { x - 5 } { x } ; x \neq - 5,0 , \frac { 3 } { 2 }$

7. $\frac { 4 x ( 2 x - 3 ) ^ { 2 } } { 2 x + 3 } ; x \neq \pm \frac { 3 } { 2 } , 0$

Exercise $\PageIndex{14}$

Perform the operations and simplify. Assume all variable expressions in the denominator are nonzero.

$\frac { a ^ { 2 } - b ^ { 2 } } { 4 a ^ { 2 } b ^ { 2 } + 4 a b ^ { 3 } } \cdot \frac { 2 a b } { a ^ { 2 } - 2 a b + b ^ { 2 } }$
$\frac { a ^ { 2 } - 5 a b + 6 b ^ { 2 } } { a ^ { 2 } - 4 a b + 4 b ^ { 2 } } \div \frac { 9 b ^ { 2 } - a ^ { 2 } } { 3 a ^ { 3 } b - 6 a ^ { 2 } b ^ { 2 } }$
$\frac { x ^ { 2 } + x y + y ^ { 2 } } { 4 x ^ { 2 } + 3 x y - y ^ { 2 } } \cdot \frac { x ^ { 2 } - y ^ { 2 } } { x ^ { 3 } - y ^ { 3 } } \div \frac { x + y } { 12 x ^ { 2 } y - 3 x y ^ { 2 } }$
$\frac { x ^ { 4 } - y ^ { 4 } } { x ^ { 2 } - 2 x y + y ^ { 2 } } \div \frac { x ^ { 2 } - 4 x y - 5 y ^ { 2 } } { 10 x ^ { 3 } } \cdot \frac { 2 x ^ { 2 } - 11 x y + 5 y ^ { 2 } } { 2 x ^ { 3 } y + 2 x y ^ { 3 } }$

Answer

1. $\frac { 1 } { 2 b ( a - b ) }$

3. $\frac { 3 x y } { x + y }$

Exercise $\PageIndex{15}$

Perform the operations and state the restrictions.

Given $f ( x ) = \frac { 4 x ^ { 2 } + 39 x - 10 } { x ^ { 2 } + 3 x - 10 }$ and $g ( x ) = \frac { 2 x ^ { 2 } + 7 x - 15 } { x ^ { 2 } + 13 x + 30 }$, find $( f \cdot g ) ( x )$.
Given $f ( x ) = \frac { 25 - x ^ { 2 } } { 3 + x }$ and $g ( x ) = \frac { 9 - x ^ { 2 } } { 5 - x }$, find $( f \cdot g ) ( x )$.
Given $f ( x ) = \frac { 42 x ^ { 2 } } { 2 x ^ { 2 } + 3 x - 2 }$ and $g ( x ) = \frac { 14 x } { 4 x ^ { 2 } - 4 x + 1 }$, find $( f / g ) ( x )$.
Given $f ( x ) = \frac { x ^ { 2 } - 20 x + 100 } { x ^ { 2 } - 1 }$ and $g ( x ) = \frac { x ^ { 2 } - 100 } { x ^ { 2 } + 2 x + 1 }$, find $( f / g ) ( x )$.
The daily cost in dollars of running a small business is given by $C (x) = 150 + 45x$ where $x$ represents the number of hours the business is in operation. Determine the average cost per hour if the business is in operation for $8$ hours in a day.
An electric bicycle manufacturer has determined that the cost of producing its product in dollars is given by the function $C (n) = 2n^{2} + 100n + 2,500$ where $n$ represents the number of electric bicycles produced in a day. Determine the average cost per bicycle if $10$ and $20$ are produced in a day.
Given $f ( x ) = 3 x - 5$, simplify $\frac { f ( x + h ) - f ( x ) } { h }$.
Given $g ( x ) = 2 x ^ { 2 } - x + 1$, simplify $\frac { g ( x + h ) - g ( x ) } { h }$.

Answer

1. $( f \cdot g ) ( x ) = \frac { ( 4 x - 1 ) ( 2 x - 3 ) } { ( x - 2 ) ( x + 3 ) } ; x \neq - 10 , - 5 , - 3,2$

3. $( f / g ) ( x ) = \frac { 3 x ( 2 x - 1 ) } { x + 2 } ; x \neq - 2,0 , \frac { 1 } { 2 }$

5. $\$ 63.75$ per hour

7. $3$

Exercise $\PageIndex{16}$

State the restrictions and simplify.

$\frac { 5 x - 6 } { x ^ { 2 } - 36 } - \frac { 4 x } { x ^ { 2 } - 36 }$
$\frac { 2 } { x } + 5 x$
$\frac { 5 } { x - 5 } + \frac { 1 } { 2 x }$
$\frac { x } { x - 2 } + \frac { 3 } { x + 3 }$
$\frac { 7 ( x - 1 ) } { 4 x ^ { 2 } - 17 x + 15 } - \frac { 2 } { x - 3 }$
$\frac { 5 } { x } - \frac { 19 x + 25 } { 2 x ^ { 2 } + 5 x }$
$\frac { x } { x - 5 } - \frac { 2 } { x - 3 } - \frac { 5 ( x - 3 ) } { x ^ { 2 } - 8 x + 15 }$
$\frac { 3 x } { 2 x - 1 } - \frac { x - 4 } { x + 4 } + \frac { 12 ( 2 - x ) } { 2 x ^ { 2 } + 7 x - 4 }$
$\frac { 1 } { t - 1 } + \frac { 1 } { ( t - 1 ) ^ { 2 } } - \frac { 1 } { t ^ { 2 } - 1 }$
$\frac { 1 } { t - 1 } - \frac { 2 t - 5 } { t ^ { 2 } - 2 t + 1 } - \frac { 5 t ^ { 2 } - 3 t - 2 } { ( t - 1 ) ^ { 3 } }$
$2 x ^ { - 1 } + x ^ { - 2 }$
$( x - 4 ) ^ { - 1 } - 2 x ^ { - 2 }$

Answer

1. $\frac { 1 } { x + 6 } ; x \neq \pm 6$

3. $\frac { 11 x - 5 } { 2 x ( x - 5 ) } ; x \neq 0,5$

5. $- \frac { 1 } { 4 x - 5 } ; x \neq \frac { 5 } { 4 } , 3$

7. $\frac { x - 5 } { x - 3 } ; x \neq 3,5$

9. $\frac { t ^ { 2 } + 1 } { ( t + 1 ) ( t - 1 ) ^ { 2 } } ; t \neq \pm 1$

11. $\frac { 2 x + 1 } { x ^ { 2 } } ; x \neq 0$

Exercise $\PageIndex{17}$

Simplify. Assume that all variable expressions used as denominators are nonzero.

$\frac { \frac { 1 } { 7 } + \frac { 1 } { x } } { \frac { 1 } { 49 } - \frac { 1 } { x ^ { 2 } } }$
$\frac { \frac { 1 } { 100 } - \frac { 1 } { x ^ { 2 } } } { \frac { 1 } { x } - \frac { 1 } { 10 } }$
$\frac { \frac { 3 } { x } - \frac { 1 } { x - 5 } } { \frac { 5 } { x + 2 } - \frac { 2 } { x } }$
$\frac { 1 - \frac { 12 } { x } + \frac { 35 } { x ^ { 2 } } } { 1 - \frac { 25 } { x ^ { 2 } } }$
$\frac { x - 4 x ^ { - 1 } } { 2 - 5 x ^ { - 1 } + 2 x ^ { - 2 } }$
$\frac { 8 x ^ { - 1 } + y ^ { - 1 } } { y ^ { - 2 } - 64 x ^ { - 2 } }$

Answer

1. $\frac { 7 x } { x - 7 }$

3. $\frac { ( x + 2 ) ( 2 x - 15 ) } { ( x - 5 ) ( 3 x - 4 ) }$

5. $\frac { x ( x + 2 ) } { 2 x - 1 }$

Exercise $\PageIndex{18}$

Perform the operations and state the restrictions.

Given $f ( x ) = \frac { 3 } { x - 3 }$ and $g ( x ) = \frac { x - 2 } { x + 2 }$, find $( f + g ) ( x )$.
Given $f ( x ) = \frac { 1 } { x ^ { 2 } + x }$ and $g ( x ) = \frac { 2 x } { x ^ { 2 } - 1 }$, find $( f + g ) ( x )$.
Given $f ( x ) = \frac { x - 3 } { x - 5 }$ and $g ( x ) = \frac { x ^ { 2 } - x } { x ^ { 2 } - 25 }$, find $( f - g ) ( x )$.
Given $f ( x ) = \frac { 11 x + 4 } { x ^ { 2 } - 2 x - 8 }$ and $g ( x ) = \frac { 2 x } { x - 4 }$, find $( f - g ) ( x )$.

Answer

1. $( f + g ) ( x ) = \frac { x ^ { 2 } - 2 x + 12 } { ( x - 3 ) ( x + 2 ) } ; x \neq - 2,3$

3. $( f - g ) ( x ) = \frac { 3 } { x + 5 } ; x \neq \pm 5$

Exercise $\PageIndex{19}$

Solve.

$\frac { 3 } { x } = \frac { 1 } { 2 x + 15 }$
$\frac { x } { x - 4 } = \frac { x + 8 } { x - 8 }$
$\frac { x + 5 } { 2 ( x + 2 ) } + \frac { x - 2 } { x + 4 } = 1$
$\frac { 2 x } { x - 5 } + \frac { 1 } { x + 1 } = 0$
$\frac { x + 1 } { x - 4 } + \frac { 4 } { x + 6 } = - \frac { 10 } { x ^ { 2 } + 2 x - 24 }$
$\frac { 2 } { x } - \frac { 12 } { 2 x + 3 } = \frac { 2 - 3 x ^ { 2 } } { 2 x ^ { 2 } + 3 x }$
$\frac { x + 7 } { x - 2 } - \frac { 9 } { x + 7 } = \frac { 81 } { x ^ { 2 } + 5 x - 14 }$
$\frac { x } { x + 5 } + \frac { 1 } { x - 4 } = \frac { 4 x - 7 } { x ^ { 2 } + x - 20 }$
$\frac { 2 } { 3 x - 1 } + \frac { x } { 2 x + 1 } = \frac { 2 ( 3 - 4 x ) } { 6 x ^ { 2 } + x - 1 }$
$\frac { x } { x - 1 } + \frac { 1 } { x + 1 } = \frac { 2 x } { x ^ { 2 } - 1 }$
$\frac { 2 x } { x + 5 } - \frac { 1 } { 2 x - 3 } = \frac { 4 - 7 x } { 2 x ^ { 2 } + 7 x - 15 }$
$\frac { x } { x + 4 } + \frac { 1 } { 2 x + 7 } = \frac { x + 8 } { 2 x ^ { 2 } + 15 x + 28 }$
$\frac { 1 } { t - 1 } - \frac { 2 } { 2 t + 1 } = \frac { 1 } { t - 2 } - \frac { 2 } { 2 t - 1 }$
$\frac { t - 1 } { t - 2 } - \frac { t - 2 } { t - 3 } = \frac { t - 3 } { t - 4 } - \frac { t - 4 } { t - 5 }$
Solve for $a$ : $\frac { 1 } { a } = \frac { 1 } { b } - \frac { 1 } { c }$
Solve for $y$ : $x = \frac { 3 y - 1 } { y - 5 }$
A positive integer is $4$ less than another. If the reciprocal of the larger integer is subtracted from twice the reciprocal of the smaller, the result is $\frac{1}{6}$. Find the two integers.
If $3$ times the reciprocal of the larger of two consecutive odd integers is added to $7$ times the reciprocal of the smaller, the result is $\frac{4}{3}$. Find the integers.
If the reciprocal of the smaller of two consecutive integers is subtracted from three times the reciprocal of the larger, the result is $\frac{3}{10}$. Find the integers.
A positive integer is twice that of another. The sum of the reciprocals of the two positive integers is $\frac{1}{4}$. Find the two integers.

Answer

1. $−9$

3. $−1, 4$

5. $−11, 0$

7. $Ø$

9. $−4$

11. $−\frac{3 }{2}$

13. $\frac{3}{4}$

15. $a = \frac { b c } { c - b }$

17. ${8, 12}$

19. ${5, 6}$

Exercise $\PageIndex{20}$

Use algebra to solve the following applications.

Manuel traveled $8$ miles on the bus and another $84$ miles on a train. If the train was $16$ miles per hour faster than the bus, and the total trip took $2$ hours, what was the average speed of the train?
A boat can average $10$ miles per hour in still water. On a trip downriver, the boat was able to travel $7.5$ miles with the current. On the return trip, the boat was only able to travel $4.5$ miles in the same amount of time against the current. What was the speed of the current?
Susan can jog, on average, $1 \frac{1}{2}$ miles per hour faster than her husband Bill. Bill can jog $10$ miles in the same amount of time it takes Susan to jog $13$ miles. How fast, on average, can Susan jog?
In the morning, Raul drove $8$ miles to visit his grandmother and then returned later that evening. Because of traffic, his average speed on the return trip was $\frac{1}{2}$ that of his average speed that morning. If the total driving time was $\frac{3}{4}$ of an hour, what was his average speed on the return trip?
One pipe can completely fill a water tank in $6$ hours while another smaller pipe takes $8$ hours to fill the same tank. How long will it take to fill the tank to $\frac{3}{4}$ capacity if both pipes are turned on?
It takes Bill $3$ minutes longer than Jerry to fill an order. Working together they can fill $15$ orders in $30$ minutes. How long does it take Bill to fill an order by himself?
Manny takes twice as long as John to assemble a skateboard. If they work together, they can assemble a skateboard in $6$ minutes. How long would it take Manny to assemble the skateboard without John’s help?
Working alone, Joe can complete the yard work in $30$ minutes. It takes Mike $45$ minutes to complete work on the same yard. How long would it take them working together?

Answer

1. $48$ miles per hour

3. $6.5$ miles per hour

5. Approximately $2.6$ hours

7. $18$ minutes

Exercise $\PageIndex{21}$

Construct a mathematical model given the following:

$y$ varies directly as $x$, where $y=30$ when $x=5$.
$y$ varies inversely as $x$, where $y=3$ when $x=-2$.
$y$ is jointly proportional to $x$ and $z$, where $y=-50$ when $x=-2$ and $z=5$.
$y$ is directly proportional to the square of $x$ and inversely proportional to $z$, where $y=-6$ when $x=2$ and $z=-8$.
The distance an object in free fall varies directly with the square of the time that it has been falling. It is observed that an object falls $36$ feet in $1 \frac{1}{2}$ seconds. Find an equation that models the distance an object will fall, and use it to determine how far it will fall in $2 \frac{1}{2}$ seconds.
After the brakes are applied, the stopping distance $d$ of an automobile varies directly with the square of the speed $s$ of the car. If a car traveling $55$ miles per hour takes $181.5$ feet to stop, how many feet will it take to stop if it is moving $65$ miles per hour?
The weight of an object varies inversely as the square of its distance from the center of the Earth. If an object weighs $180$ lbs on the surface of the Earth (approximately $4,000$ miles from the center), then how much will it weigh at $2,000$ miles above the Earth’s surface?
The cost per person of renting a limousine varies inversely with the number of people renting it. If $5$ people go in on the rental, the limousine will cost $$112$ per person. How much will the rental cost per person if $8$ people go in on the rental?
To balance a seesaw, the distance from the fulcrum that a person must sit is inversely proportional to his weight. If a $52$-pound boy is sitting $3$ feet away from the fulcrum, then how far from the fulcrum must a $44$-pound boy sit? Round to the nearest tenth of a foot.

Answer

1. $y=6x$

3. $y=5xz$

5. $d=16t^{2}$; $100$ feet

7. $80$ lbs

9. Approximately $3.5$ feet

Sample Exam

Exercise $\PageIndex{22}$

Given $f ( x ) = x ^ { 2 } - x + 4 , g ( x ) = 5 x - 1$, and $h ( x ) = 2 x ^ { 2 } + x - 3$, find the following:

$( g \cdot h ) ( x )$
$( h - f ) ( x )$
$( f + g ) ( - 1 )$

Answer

1. $( g \cdot h ) ( x ) = 10 x ^ { 3 } + 3 x ^ { 2 } - 16 x + 3$

3. $( f + g ) ( - 1 ) = 0$

Exercise $\PageIndex{23}$

Factor.

$x ^ { 3 } + 16 x - 2 x ^ { 2 } - 32$
$x ^ { 3 } - 8 y ^ { 3 }$
$x ^ { 4 } - 81$
$25 x ^ { 2 } y ^ { 2 } - 40 x y + 16$
$16 x ^ { 3 } y + 12 x ^ { 2 } y ^ { 2 } - 18 x y ^ { 3 }$

Answer

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

4. $( 5 x y - 4 ) ^ { 2 }$

Exercise $\PageIndex{24}$

Solve

$6 x ^ { 2 } + 24 x = 0$
$( 2 x + 1 ) ( 3 x + 2 ) = 12$
$( 2 x + 1 ) ^ { 2 } = 23 x + 6$
Find a quadratic equation with integer coefficients given the solutions $\left\{ \frac { 1 } { 2 } , - 3 \right\}$.
Given $f ( x ) = 5 x ^ { 2 } - x + 4$, simplify $\frac { f ( x + h ) - f ( x ) } { h }$, where $h \neq 0$.

Answer

1. $-4,0$

3. $-\frac{1}{4}, 5$

5. $10x+5h-1$

Exercise $\PageIndex{25}$

Simplify and state the restrictions.

$\frac { 4 x ^ { 2 } - 33 x + 8 } { x ^ { 2 } - 10 x + 16 } \div \frac { 16 x ^ { 2 } - 1 } { x ^ { 2 } - 4 x + 4 }$
$\frac { x - 1 } { x - 7 } + \frac { 1 } { 1 - x } - \frac { 2 ( x + 11 ) } { x ^ { 2 } - 8 x + 7 }$

Answer: 2. $\frac { x + 2 } { x - 1 } ; x \neq 1,7$

Exercise $\PageIndex{26}$

Assume all variable expressions in the denominator are nonzero and simplify.

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

Exercise $\PageIndex{27}$

Solve.

$\frac { 6 x - 5 } { 3 x + 2 } = \frac { 2 x } { x + 1 }$
$\frac { 2 x } { x + 5 } - \frac { 1 } { 5 - x } = \frac { 2 x } { x ^ { 2 } - 25 }$
Find the root of the function defined by $f ( x ) = \frac { 1 } { x + 3 } - 4$.
Solve for $y$: $x = \frac { 4 y } { 3 y - 1 }$

Answer

1. $-\frac{5}{3}$

3. $-\frac{11}{4}$

Exercise $\PageIndex{28}$

Use algebra to solve.

The height of an object dropped from a $64$-foot building is given by the function $h (t) = −16t^{2} + 64$, where $t$ represents time in seconds after it was dropped.
1. Determine the height of the object at $\frac{3}{4}$ of a second.
2. How long will it take the object to hit the ground?
One positive integer is $3$ units more than another. When the reciprocal of the larger is subtracted from twice the reciprocal of the smaller, the result is $\frac{2}{9}$. Find the two positive integers.
A light airplane can average $126$ miles per hour in still air. On a trip, the airplane traveled $222$ miles with a tailwind. On the return trip, against a headwind of the same speed, the plane was only able to travel $156$ miles in the same amount of time. What was the speed of the wind?
On the production line, it takes John $2$ minutes less time than Mark to assemble a watch. Working together they can assemble $5$ watches in $12$ minutes. How long does it take John to assemble a watch working alone?
Write an equation that relates $x$ and $y$, given that $y$ varies inversely with the square of $x$, where $y = −\frac{1}{3}$ when $x = 3$. Use it to find $y$ when $x = \frac{1}{2}$.

Answer

1. (1) $55$ feet; (2) $2$ seconds

3. $22$ miles per hour

5. $y = - \frac { 3 } { x ^ { 2 } } ; y = - 12$

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