Chapter 5

18 plus 11 OR the sum of 18 and 11

27 times 9 OR the product of 27 and 9

84 divided by 7 OR the quotient of 84 and 7

$p$ minus $q$ OR the difference of $p$ and $q$

$43+67$

$\left(45\right)\left(3\right)$

$45÷<!--\; ÷\; -->3$

$89-<!--\; -\; -->42$

$n-<!--\; -\; -->20$

$\left(n+2\right)\bullet <!--\; \bullet \; -->6$ OR $6\left(n+2\right)$

$n^3-<!--\; -\; -->5$

$60h+40$

$5y=50$

$\frac{1}{2}n=30$

$3n-<!--\; -\; -->7=2$

$2x+7=21$

$24-<!--\; -\; -->\left(17-<!--\; -\; -->6\right)=13$

$\left(3\bullet <!--\; \bullet \; -->6\right)+13=31$

$\left(12-<!--\; -\; -->6\right)÷<!--\; ÷\; -->\left(5-<!--\; -\; -->3\right)=3$

$5\bullet <!--\; \bullet \; -->\left(3^2+5\right)=70$

$9$

$-<!--\; -\; -->6$

$5x^2-<!--\; -\; -->3x-<!--\; -\; -->7$

$-<!--\; -\; -->9x–4$

$2y+10$

$-<!--\; -\; -->2a-<!--\; -\; -->2b+8$

$112+16x$

$6x+18$

$3a+4$

$2x^2-<!--\; -\; -->11x+12$

$4x^2+x-<!--\; -\; -->2$

$x=3$

$y=28$

Henry has 42 books.

Total cost is $48.85

Answers will vary. For example: You can rent a paddleboard for $25 per hour with a water shoe purchase of $75. If you spent $200, how many hours did you rent the paddle board for?
You rented the paddle board for 5 hours.

$12=-<!--\; -\; -->1$, which is false; therefore, this is a false statement, and the equation has no solution.

$-<!--\; -\; -->12=-<!--\; -\; -->12$, which is true; therefore, this is a true statement, and there are infinitely many solutions.

$t=\frac{I}{\text{P}r}$

$h=\frac{3V}{\mathrm{\pi <!--\; \pi \; -->}{r}^2}$

$0\le <!--\; \le \; -->x\le <!--\; \le \; -->2.5$
Graph:

 
$p\ge <!--\; \ge \; -->2$

Taleisha can send/receive 106 or fewer text messages and keep her monthly bill no more than $50.

You could take up to 17 credit hours and stay under $2,000.

Malik must tutor at least 23 hours.

$a$ = 1 U.S. dollar, and $b$ = 1.21 Canadian dollars, the ratio is 1 to 1.21; or 1:1.21; or $\frac{1}{1.21}$ .

With $a$ = 170 pounds on Earth, and $b$ = 64 pounds on Mars, the ratio is 170 to 64; or 170:64; or $\frac{170}{64}$.

$219.51

501.6 pounds

720 cookies

The constant of proportionality (centimeters divided by inches) is 2.54. This tells you that there are 2.54 centimeters in one inch.

30.5 hours (or 30½ hours, or 30 hours and 30 minutes)

$207.50

125.9 miles

The scale is $1\text{ inch}=91.25\text{ miles}$. The other borders would calculate as: eastern and western borders are 273.75 miles, and northern border is 365 miles.

184 inches, or 15 feet, 4 inches.

1. Yes
2. Yes

1. No
2. No

1. Yes
2. Yes

1. Yes
2. Yes

Yes

Yes

No

No

No

$y\ge <!--\; \ge \; -->-<!--\; -\; -->2x+3$

$11x+16.5y\ge <!--\; \ge \; -->330$

Answers will vary.

$x^2+4x+3$

$2x^2-<!--\; -\; -->5x-<!--\; -\; -->3$

$(x+2)(x+4)$

$\left(x-<!--\; -\; -->7\right)\left(x-<!--\; -\; -->9\right)$

$x=-<!--\; -\; -->3,x=2$

$x=2$ or $x=-<!--\; -\; -->5$

$x=3$ or $x=-<!--\; -\; -->5$

$x=5$, $x=-<!--\; -\; -->5$

$p=\frac{7}{5}$, $p=-<!--\; -\; -->\frac{7}{5}$

$a=5$, $a=-<!--\; -\; -->3$

$y=1$, $y=\frac{2}{3}$

16, 15 and –16, –15

$\text{width}=5\text{ feet}$, $\text{length}=6\text{ feet}$

22

6

$3t^2-<!--\; -\; -->2t+1$

There 205 unread emails after 7 days.

This relation is a function.

This relation is not a function.

Both George and Mike have two phone numbers. Each $x$-value is not matched with only one $y$-value. This relation is not a function.

function

not a function

function

This graph represents a function.

This graph does not represent a function.

The domain is the set of all $x$-values of the relation: $\left\{1,2,3,4,5\right\}$.

The range is the set of all $y$-values of the relation: $\left\{1,8,27,64,125\right\}$.

The ordered pairs of the relation are: $\left\{\left(-<!--\; -\; -->3,3\right),\left(-<!--\; -\; -->2,2\right),\left(-<!--\; -\; -->1,0\right),\left(0,-<!--\; -\; -->1\right),\left(2,-<!--\; -\; -->2\right),\left(4,-<!--\; -\; -->4\right)\right\}$.

The domain is the set of all $x$-values of the relation: $\left\{-<!--\; -\; -->3,-<!--\; -\; -->2,-<!--\; -\; -->1,0,2,4\right\}$.

The range is the set of all $y$-values of the relation: $\left\{-<!--\; -\; -->4,-<!--\; -\; -->2,-<!--\; -\; -->1,0,2,3\right\}$.

The graph crosses the $x$-axis at the point (2, 0). The $x$-intercept is (2, 0). The graph crosses the $y$-axis at the point (0, −2). The $y$-intercept is (0, −2).

The $x$-intercept is (4, 0) and the $y$-intercept is (0, 12).

$m=\frac{-<!--\; -\; -->4}{3}$

$\text{slope}=-<!--\; -\; -->1$

slope $m=2$ and $y\text{-intercept}=(0,-<!--\; -\; -->1)$

slope $m=-<!--\; -\; -->\frac{1}{4}$ and $y\text{-intercept}=(0,4)$

(0, 20) is the $y\text{-intercept}$ and represents that there were 20 teachers at Jones High School in 1990. There is no $x\text{-intercept}$.

In the first 5 years the slope is 2; this means that on average, the school gained 2 teachers every year between 1990 and 1995. Between 1995 and 2000, the slope is 4; on average the school gained 4 teachers every year. Then the slope is 0 between 2000 and 2005 meaning the number of teachers remained the same. There was a decrease in teachers between 2005 and 2010, represented by a slope of –2. Finally, the slope is 4 between 2010 and 2020, which indicates that on average the school gained 4 teachers every year.

Answers will vary. Jones High School was founded in 1990 and hired 2 teachers per year until 1995, when they had an increase in students and they hired 4 teachers per year for the next 5 years. Then there was a hiring freeze, and no teachers were hired between 2000 and 2005. After the hiring freeze, the student population decreased, and they lost 2 teachers per year until 2010. Another surge in student population meant Jones High School hired 4 new teachers per year until 2020 when they had 80 teachers at the school.

50 inches

66 inches

The slope, 2, means that the height $h$ increases 2 inches when the shoe size(s) increases 1 size.

The $h$-intercept means that when the shoe size is 0, the woman’s height is 50 inches.

$25

$85

The slope, 4, means that the weekly cost, $C$, increases by $4 when the number of pizzas sold, $p$, increases by 1. The $C$-intercept means that when the number of pizzas sold is 0, the weekly cost is $25.

No

Yes

No

No

(3, 2)

$(3,2)$

There is no solution to this system.

There are infinitely many solutions to this system.

Jenna burns 8.3 calories per minute circuit training and 11.2 calories per minute while on the elliptical trainer.

not a solution

a solution

The region containing (0, 0) is the solution to the system of linear inequalities.

The solution is the darkest shaded region.

The solution is the lighter shaded region.

No solution.

The solution is the lighter shaded region.

$\{\begin{array}{l}240h+160C\ge <!--\; \ge \; -->800\\ 1.40h+0.50C\le <!--\; \le \; -->5\\ h\ge <!--\; \ge \; -->0\\ C\ge <!--\; \ge \; -->0\end{array}$

The point $(3,2)$ is not in the solution region. Omar would not choose to eat 3 hamburgers and 2 cookies.

The point $(2,4)$ is in the solution region. Omar might choose to eat 2 hamburgers and 4 cookies.

With $a=$ the number of bags of apples sold, and $b=$ the number of bunches of bananas sold, the objective function is $P=4a+6b$.

$T=20x+28y$

The constraints are $a+b\le <!--\; \le \; -->20$ and $3a+5b\le <!--\; \le \; -->70$. The summary is: $P=4a+6b$, $a+b\le <!--\; \le \; -->20$, and $3a+5b\le <!--\; \le \; -->70$.

The constraints are:
$15\le <!--\; \le \; -->x\le <!--\; \le \; -->22$
$13\le <!--\; \le \; -->y\le <!--\; \le \; -->19$
So the system is:
$T=20x+28y$
$15\le <!--\; \le \; -->x\le <!--\; \le \; -->22$
$13\le <!--\; \le \; -->y\le <!--\; \le \; -->19$

The maximum value for the profit $P$ occurs when $x=15$ and $y=5$. This means that to maximize their profit, the Robotics Club should sell 15 bags of apples and 5 bunches of bananas every day.

$J=V+2$, $V=J-<!--\; -\; -->2$

$5x+8$, $2n+3m$

$(8x+12x)÷<!--\; ÷\; -->\left(4x-<!--\; -\; -->2x\right)$

$-<!--\; -\; -->3$

$x^2-<!--\; -\; -->2xy+y^2$

$3x\left(3x^2+x-<!--\; -\; -->2\right)$

multiplication

division

It is a correct solution strategy.
Let

It is a correct solution strategy.
Let

This is not a correct solution strategy. The negative sign is not distributed correctly in the second line of the solution strategy. The second line should read $8x+7-<!--\; -\; -->2x+9=22-<!--\; -\; -->4x+4$.

$T=1.7x+3$

$40.40

$T=1.6y+5$

$40.20

The Enjoyable Cab Company, because the cab fare will be $ 0.20 less than what it would cost to take a taxi from the Nice Cab Company.

Luis is; there are infinitely many solutions. If this is solved using the general strategy, it simplifies to $0=0$. This is a true statement, so there are infinitely many solutions.

d $F=\frac{9}{5}C+32$

b $K=C+273$, although d $C=K–273$ is an equivalent formula.

a $K=\frac{5}{9}\left(F-<!--\; -\; -->32\right)+273$

b $R=\frac{9}{5}C+492$

d $\left(-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},-<!--\; -\; -->1\right)$

b $[-<!--\; -\; -->5,\mathrm{\infty <!--\; \infty \; -->})$

c $\left[\frac{3}{2},\mathrm{\infty <!--\; \infty \; -->}\right)$

a $(-<!--\; -\; -->4,3)$

e $6x\ge <!--\; \ge \; -->24$

b $-<!--\; -\; -->6x><!--\; >\; -->18$

c $4x-<!--\; -\; -->3>-<!--\; -\; -->11$

c $-<!--\; -\; -->3x+14>-<!--\; -\; -->13$

b

d $50x>\text{8,120}$

a True

a True

a True

b False

a True

$12:16$

$16:28$

324.00 British pounds (None of these.)

10 inches

Yes he can, but barely. At 37 miles per gallon, Albert can drive $499.5$ miles. While in theory he can make it, he probably should fill up with gasoline somewhere along the way!

$\$40.66$

d $y=\frac{5}{3}$

b $y=\frac{1}{2}x+4$

b $y\le <!--\; \le \; -->-<!--\; -\; -->2x+5$

d

b

c

d

b False

b False

a True

b False

$x=\frac{5}{2}\pm <!--\; \pm \; -->\frac{\sqrt{5}}{2}$

a True

b False

b False

a True

b False

a True

b False

b False

a True

b False

Elimination

Substitution

Substitution

Elimination

Substitution

Elimination

Elimination

Substitution

c

e

d

b

a

c

a

b

d

a

c

d

b

$\$140$