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Using the paper provided, writing/coloring utensils, and drawing directions, you are to create a new map of Parallel City. Directions: Draw the buildings on the appropriate corners on the Parallel City $2$ sheet provided (you may want to practice on the back of this sheet FIRST). Make it neat, label it, and be creative. Construction: $1$ . The Bank and Wal-Mart are alternate interior angles. $2$ . A Gas Station and the Post Office are vertical angles. $3$ . Your house and the Post Office are consecutive interior angles. $4$ . Your house and the Park are alternate interior angles. $5$ . A Gas Station and the Hospital are a pair of supplementary angles. $6$ . The Church and the Hospital are alternate exterior angles.

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One-step inequalities: word problems

Full solution

Q. Using the paper provided, writing/coloring utensils, and drawing directions, you are to create a new map of Parallel City. Directions: Draw the buildings on the appropriate corners on the Parallel City

2

sheet provided (you may want to practice on the back of this sheet FIRST). Make it neat, label it, and be creative. Construction:

1

. The Bank and Wal-Mart are alternate interior angles.

2

. A Gas Station and the Post Office are vertical angles.

3

. Your house and the Post Office are consecutive interior angles.

4

. Your house and the Park are alternate interior angles.

5

. A Gas Station and the Hospital are a pair of supplementary angles.

6

. The Church and the Hospital are alternate exterior angles.

Draw Bank and Wal-Mart: Draw the Bank and Wal-Mart as alternate interior angles on the map. This means they should be on opposite sides of a street but not directly across from each other.
Place Gas Station and Post Office: Place the Gas Station and the Post Office as vertical angles, which means they should be across the street from each other at an intersection.
House and Post Office: Your house and the Post Office are consecutive interior angles, so they should be next to each other on the same side of a street.
House and Park: Your house and the Park are alternate interior angles, so they should be on opposite sides of a street but not directly across from each other.
Gas Station and Hospital: The Gas Station and the Hospital are a pair of supplementary angles, which means they should form a straight line when combined. Place them on opposite corners of an intersection.
Church and Hospital: The Church and the Hospital are alternate exterior angles, so they should be on opposite sides of a street but not facing each other or adjacent.

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-11 x \geq-33

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x \leq-3

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x \geq-3

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x \leq 3

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\frac{3}{2} q \leq \frac{9}{2} q-18

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q \leq 3

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Question

g(x)=5-e^{x}

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Below is Sean's attempt to write a formal justification for the fact that the equation

g(x)=0

has a solution where

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Is Sean's justification complete? If not, why?

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Sean's justification:

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g

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So, according to the intermediate value theorem,

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must have a solution somewhere between

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(A) Yes, Sean's justification is complete.

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(B) No, Sean didn't establish that

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Question

f(x)=\sqrt[3]{\cos (\pi \cdot x)}

\newline

Below is Issac's attempt to write a formal justification for the fact that the equation

f(x)=0.1

has a solution where

1.5 \leq x \leq 2

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Is Issac's justification complete? If not, why?

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Issac's justification:

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\begin{array}{l} f(1.5)=0 \text { and } \\ f(2)=1 \text {, so } \\ f(1.5)<0.1<f(2) . \end{array}

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So, according to the intermediate value theorem,

f(x)=0.1

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x

x=1.5

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(A) Yes, Issac's justification is complete.

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(B) No, Issac didn't establish that

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f(1.5)

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(C) No, Issac didn't establish that

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Question

h(x)=5 \cdot \tan (x)

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Below is Bridget's attempt to write a formal justification for the fact that the equation

h(x)=2

has a solution where

0 \leq x \leq \frac{\pi}{4}

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Is Bridget's justification complete? If not, why?

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Bridget's justification:

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h

is defined over the entire interval

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So, according to the intermediate value theorem,

h(x)=2

must have a solution at some point in that interval.

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(A) Yes, Bridget's justification is complete.

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(B) No, Bridget didn't establish that

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h(0)

h\left(\frac{\pi}{4}\right)

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(C) No, Bridget didn't establish that

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Question

f(x)=2^{5-2 x}

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Below is Ibrahim's attempt to write a formal justification for the fact that the equation

f(x)=10

has a solution where

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Is Ibrahim's justification complete? If not, why?

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Ibrahim's justification:

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f

is defined for all real numbers, and exponential functions are continuous at all points in their domains. Also,

f(-1)=128

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f(-1)

f(4)

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So, according to the intermediate value theorem,

f(x)=10

must have a solution at some point between

x=-1

x=4

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1

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(A) Yes, Ibrahim's justification is complete.

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(B) No, Ibrahim didn't establish that

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(c) No, Ibrahim didn't establish that

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Question

g(x)=\cos \left(\pi x^{2}\right)

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Below is Amrita's attempt to write a formal justification for the fact that the equation

g^{\prime}(x)=0.4

has a solution where

-1<x<4

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Is Amrita's justification complete?

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Amrita's justification:

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Polynomial and trigonometric functions are differentiable and continuous at all points in their domain, and

[-1,4]

g^{\prime}

s domain. Also,

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\begin{array}{l} g(-1)=-1 \text { and } \\ g(4)=1 \text {, so } \\ \frac{g(4)-g(-1)}{4-(-1)}=0.4 . \end{array}

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So, according to the mean value theorem,

g^{\prime}(x)=0.4

must have a solution somewhere between

x=-1

x=4

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1

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(A) Yes, Amrita's justification is complete.

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(B) No, Amrita didn't establish that the average rate of change of

g

[-1,4]

0

4

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(C) No, Amrita didn't establish that

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Question

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h(x)=x^{5}-2 x^{4}+10 x^{2}-10 x \text {. }

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Below is Omar's attempt to write a formal justification for the fact that there exists a value

c

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-2<c<2 \text { such that } h^{\prime}(c)=6 \text {. }

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Is Omar's justification complete? If not, why?

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Omar's justification:

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\begin{array}{l} h(-2)=-4 \text { and } \\ h(2)=20 \text {, so } \\ \frac{h(2)-h(-2)}{2-(-2)}=6 . \end{array}

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c

somewhere in the interval

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(A) Yes, Omar's justification is complete.

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(B) No, Omar didn't establish that the average rate of change of

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Question

f(x)=\sqrt{x+\sqrt{x^{3}}}

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Below is Lena's attempt to write a formal justification for the fact that the equation

f^{\prime}(x)=\frac{2}{3}

has a solution where

0<x<9

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Is Lena's justification complete? If not, why?

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Lena's justification:

\newline

f(0)=0

f(9)=6

, so the average rate of change of

f

over the interval

0 \leq x \leq 9

\frac{2}{3}

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So, according to the mean value theorem,

f^{\prime}(x)=\frac{2}{3}

must have a solution somewhere in the interval

0<x<9

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1

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(A) Yes, Lena's justification is complete.

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(B) No, Lena didn't establish that the average rate of change of

f

[0,9]

\frac{2}{3}

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(C) No, Lena didn't establish that

f

is differentiable.

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Question

h(x)=3^{x}-x^{2}

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Below is Uriah's attempt to write a formal justification for the fact that the equation

h^{\prime}(x)=8

has a solution where

1<x<3

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Is Uriah's justification complete? If not, why?

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Uriah's justification:

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Polynomial and exponential functions are differentiable and continuous at all points in their domain, and

[1,3]

h

's domain. Also,

h(1)=2

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\frac{h(3)-h(1)}{3-1}=8 \text {. }

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So, according to the mean value theorem,

h^{\prime}(x)=8

must have a solution somewhere between

x=1

x=3

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1

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(A) Yes, Uriah's justification is complete.

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(B) No, Uriah didn't establish that the average rate of change of

h

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8

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(C) No, Uriah didn't establish that

h

is differentiable.

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