Algebra and Trigonometry

Jay Abramson

Chapter 2

Equations and Inequalities - all with Video Answers

Educators

Section 1

The Rectangular Coordinate Systems and Graphs

Problem 1

Is it possible for a point plotted in the Cartesian coordinate system to not lie in one of the four quadrants? Explain.

Zach Steedman

Zach Steedman

Numerade Educator

Problem 2

Describe the process for finding the $x$ -intercept and the $y$ -intercept of a graph algebraically.

Rishi Kavikondala

Rishi Kavikondala

Numerade Educator

Problem 3

Describe in your own words what the $y$ -intercept of a graph is.

Zach Steedman

Numerade Educator

Problem 4

When using the distance formula $d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}},$ explain the correct order of operations that are to be performed to obtain the correct answer.

Rishi Kavikondala

Rishi Kavikondala

Numerade Educator

Problem 5

Find the $x$ -intercept and the $y$ -intercept without graphing. Write the coordinates of each intercept.
$$y=-3 x+6$$

Julie Silva

Numerade Educator

Problem 6

Find the $x$ -intercept and the $y$ -intercept without graphing. Write the coordinates of each intercept.
$$4 y=2 x-1$$

Julie Silva

Numerade Educator

Problem 7

Find the $x$ -intercept and the $y$ -intercept without graphing. Write the coordinates of each intercept.
$$3 x-2 y=6$$

Breanna Ollech

Numerade Educator

Problem 8

Find the $x$ -intercept and the $y$ -intercept without graphing. Write the coordinates of each intercept.
$$4 x-3=2 y$$

Julie Silva

Numerade Educator

Problem 9

Find the $x$ -intercept and the $y$ -intercept without graphing. Write the coordinates of each intercept.
$$3 x+8 y=9$$

Julie Silva

Numerade Educator

Problem 10

Find the $x$ -intercept and the $y$ -intercept without graphing. Write the coordinates of each intercept.
$$2 x-\frac{2}{3}=\frac{3}{4} y+3$$

Julie Silva

Numerade Educator

Problem 11

Solve the equation for $y$ in terms of $x$
$$4 x+2 y=8$$

Julie Silva

Numerade Educator

Problem 12

Solve the equation for $y$ in terms of $x$
$$3 x-2 y=6$$

Julie Silva

Numerade Educator

Problem 13

Solve the equation for $y$ in terms of $x$
$$2 x=5-3 y$$

Julie Silva

Numerade Educator

Problem 14

Solve the equation for $y$ in terms of $x$
$$x-2 y=7$$

Julie Silva

Numerade Educator

Problem 15

Solve the equation for $y$ in terms of $x$
$$5 y+4=10 x$$

Julie Silva

Numerade Educator

Problem 16

Solve the equation for $y$ in terms of $x$
$$5 x+2 y=0$$

Julie Silva

Numerade Educator

Problem 17

Find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.
$$(-4,1) \text { and }(3,-4)$$

Julie Silva

Numerade Educator

Problem 18

Find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.
$$(2,-5) \text { and }(7,4)$$

Julie Silva

Numerade Educator

Problem 19

Find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.
$$(5,0) \text { and }(5,6)$$

Julie Silva

Numerade Educator

Problem 20

Find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.
$$(-4,3) \text { and }(10,3)$$

Julie Silva

Numerade Educator

Problem 21

Find the distance between the two points given using your calculator, and round your answer to the nearest hundredth. (19, 12) and ( 41,71 )

Julie Silva

Numerade Educator

Problem 22

Find the coordinates of the midpoint of the line segment that joins the two given points.
$$(-5,-6) \text { and }(4,2)$$

Breanna Ollech

Numerade Educator

Problem 23

Find the coordinates of the midpoint of the line segment that joins the two given points.
$$(-1,1) \text { and }(7,-4)$$

Julie Silva

Numerade Educator

Problem 24

Find the coordinates of the midpoint of the line segment that joins the two given points.
$$(-5,-3) \text { and }(-2,-8)$$

Julie Silva

Numerade Educator

Problem 25

Find the coordinates of the midpoint of the line segment that joins the two given points.
$$(0,7) \text { and }(4,-9)$$

Julie Silva

Numerade Educator

Problem 26

Find the coordinates of the midpoint of the line segment that joins the two given points.
$$(-43,17) \text { and }(23,-34)$$

Julie Silva

Numerade Educator

Problem 27

Identify the information requested.
What are the coordinates of the origin?

Julie Silva

Numerade Educator

Problem 28

Identify the information requested.
If a point is located on the $y$ -axis, what is the $x$ -coordinate?

Julie Silva

Numerade Educator

Problem 29

Identify the information requested.
If a point is located on the $x$ -axis, what is the $y$ -coordinate?

Julie Silva

Numerade Educator

Problem 30

Plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).
$$(4,1)(-2,-3)(5,0)$$

Julie Silva

Numerade Educator

Problem 31

Plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).
$$(-1,2)(0,4)(2,1)$$

Julie Silva

Numerade Educator

Problem 32

Plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).
$$(-3,0)(-3,4)(-3,-3)$$

Julie Silva

Numerade Educator

Problem 33

Plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).
Name the coordinates of the points graphed.

Julie Silva

Numerade Educator

Problem 34

Name the quadrant in which the following points would be located. If the point is on an axis, name the axis.
a. (-3,-4)
b. (-5,0)
c. (1,-4)
d. (-2,7)
e. (0,-3)

Rishi Kavikondala

Rishi Kavikondala

Numerade Educator

Problem 35

Construct a table and graph the equation by plotting at least three points.
$$y=\frac{1}{3} x+2$$

Julie Silva

Numerade Educator

Problem 36

Construct a table and graph the equation by plotting at least three points.
$$y=-3 x+1$$

Maria Dearborn

Numerade Educator

Problem 37

Construct a table and graph the equation by plotting at least three points.
$$2 y=x+3$$

Julie Silva

Numerade Educator

Problem 38

Find and plot the $x$ -and $y$ -intercepts, and graph the straight line based on those two points.
$$4 x-3 y=12$$

Julie Silva

Numerade Educator

Problem 39

Find and plot the $x$ -and $y$ -intercepts, and graph the straight line based on those two points.
$$x-2 y=8$$

Julie Silva

Numerade Educator

Problem 40

Find and plot the $x$ -and $y$ -intercepts, and graph the straight line based on those two points.

Julie Silva

Numerade Educator

Problem 41

Find and plot the $x$ -and $y$ -intercepts, and graph the straight line based on those two points.
$$3 y=-2 x+6$$

Julie Silva

Numerade Educator

Problem 42

Find and plot the $x$ -and $y$ -intercepts, and graph the straight line based on those two points.
$$y=\frac{x-3}{2}$$

Julie Silva

Numerade Educator

Problem 43

Use the graph in the figure below.
Find the distance between the two endpoints using the distance formula. Round to three decimal places.

Julie Silva

Numerade Educator

Problem 44

Use the graph in the figure below.
Find the coordinates of the midpoint of the line segment connecting the two points.

Julie Silva

Numerade Educator

Problem 45

Use the graph in the figure below.
Find the distance that (-3,4) is from the origin.

Julie Silva

Numerade Educator

Problem 46

Use the graph in the figure below.
Find the distance that (5,2) is from the origin. Round to three decimal places.

Julie Silva

Numerade Educator

Problem 47

Use the graph in the figure below.
Which point is closer to the origin?

Julie Silva

Numerade Educator

Problem 48

Use your graphing calculator to input the linear graphs in the Y= graph menu. After graphing it, use the $2^{\text {nd }}$ CALC button and 1:value button, hit ENTER. At the lower part of the screen you will see "x=" and a blinking cursor. You may enter any number for $x$ and it will display the $y$ value for any $x$ value you input. Use this and plug in $x=0,$ thus finding the $y$ -intercept, for each of the following graphs.
$$Y_{1}=-2 x+5$$

Julie Silva

Numerade Educator

Problem 49

Use your graphing calculator to input the linear graphs in the Y= graph menu. After graphing it, use the $2^{\text {nd }}$ CALC button and 1:value button, hit ENTER. At the lower part of the screen you will see "x=" and a blinking cursor. You may enter any number for $x$ and it will display the $y$ value for any $x$ value you input. Use this and plug in $x=0,$ thus finding the $y$ -intercept, for each of the following graphs.
$$Y_{1}=\frac{3 x-8}{4}$$

Julie Silva

Numerade Educator

Problem 50

Use your graphing calculator to input the linear graphs in the Y= graph menu. After graphing it, use the $2^{\text {nd }}$ CALC button and 1:value button, hit ENTER. At the lower part of the screen you will see "x=" and a blinking cursor. You may enter any number for $x$ and it will display the $y$ value for any $x$ value you input. Use this and plug in $x=0,$ thus finding the $y$ -intercept, for each of the following graphs.
$$Y_{1}=\frac{x+5}{2}$$

Julie Silva

Numerade Educator

Problem 51

Use your graphing calculator to input the linear graphs in the $Y=$ graph menu. After graphing it, use the $2^{\text {ad }}$ CALC button and 2 -zero button, hit ENTER. At the lower part of the screen you will see "left bound?" and a blinking cursor on the graph of the line. Move this cursor to the left of the $x$ -intercept, hit ENTER. Now it says "right bound?" Move the cursor to the right of the $x$ -intercept, hit ENTER. Now it says "guess?" Move your cursor to the left somewhere in between the left and right bound near the $x$ -intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the $x$ -intercept or the "zero" to the $y$ -value. Use this to find the $x$ -intercept.
Note: With linear/straight line functions the zero is not really a "guess," but it is necessary to enter a "guess" so it will search and find the exact $x$ -intercept between your right and left boundaries. With other types of functions (more than one $x$ -intercept), they may be irrational numbers so "guess" is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.
$$Y_{1}=-8 x+6$$

Julie Silva

Numerade Educator

Problem 52

Use your graphing calculator to input the linear graphs in the $Y=$ graph menu. After graphing it, use the $2^{\text {ad }}$ CALC button and 2 -zero button, hit ENTER. At the lower part of the screen you will see "left bound?" and a blinking cursor on the graph of the line. Move this cursor to the left of the $x$ -intercept, hit ENTER. Now it says "right bound?" Move the cursor to the right of the $x$ -intercept, hit ENTER. Now it says "guess?" Move your cursor to the left somewhere in between the left and right bound near the $x$ -intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the $x$ -intercept or the "zero" to the $y$ -value. Use this to find the $x$ -intercept.
Note: With linear/straight line functions the zero is not really a "guess," but it is necessary to enter a "guess" so it will search and find the exact $x$ -intercept between your right and left boundaries. With other types of functions (more than one $x$ -intercept), they may be irrational numbers so "guess" is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.
$$Y_{1}=4 x-7$$

Julie Silva

Numerade Educator

Problem 53

Use your graphing calculator to input the linear graphs in the $Y=$ graph menu. After graphing it, use the $2^{\text {ad }}$ CALC button and 2 -zero button, hit ENTER. At the lower part of the screen you will see "left bound?" and a blinking cursor on the graph of the line. Move this cursor to the left of the $x$ -intercept, hit ENTER. Now it says "right bound?" Move the cursor to the right of the $x$ -intercept, hit ENTER. Now it says "guess?" Move your cursor to the left somewhere in between the left and right bound near the $x$ -intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the $x$ -intercept or the "zero" to the $y$ -value. Use this to find the $x$ -intercept.
Note: With linear/straight line functions the zero is not really a "guess," but it is necessary to enter a "guess" so it will search and find the exact $x$ -intercept between your right and left boundaries. With other types of functions (more than one $x$ -intercept), they may be irrational numbers so "guess" is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.
$\mathrm{Y}_{1}=\frac{3 x+5}{4}$ Round your answer to the nearest thousandth.

Julie Silva

Numerade Educator

Problem 54

A man drove $10 \mathrm{mi}$ directly east from his home, made a left turn at an intersection, and then traveled 5 mi north to his place of work. If a road was made directly from his home to his place of work, what would its distance be to the nearest tenth of a mile?

Rishi Kavikondala

Rishi Kavikondala

Numerade Educator

Problem 55

If the road was made in the previous exercise, how much shorter would the man's one-way trip be every day?

Julie Silva

Numerade Educator

Problem 56

Given these four points: $A(1,3), B(-3,5), C(4,7)$ and $D(5,-4)$, find the coordinates of the midpoint of line segments $\overline{A B}$ and $\overline{C D}$.

Rishi Kavikondala

Rishi Kavikondala

Numerade Educator

Problem 57

After finding the two midpoints in the previous exercise, find the distance between the two midpoints to the nearest thousandth.

Zach Steedman

Numerade Educator

Problem 58

Given the graph of the rectangle shown and the coordinates of its vertices, prove that the diagonals of the rectangle are of equal length.

Rishi Kavikondala

Rishi Kavikondala

Numerade Educator

Problem 59

In the previous exercise, find the coordinates of the midpoint for each diagonal.

Zach Steedman

Numerade Educator

Problem 60

The coordinates on a map for San Francisco are (53,17) and those for Sacramento are (123,78) Note that coordinates represent miles. Find the distance between the cities to the nearest mile.

Rishi Kavikondala

Rishi Kavikondala

Numerade Educator

Problem 61

If San Jose's coordinates are $(76,-12),$ where the coordinates represent miles, find the distance between San Jose and San Francisco to the nearest mile.

Julie Silva

Numerade Educator

Problem 62

A small craft in Lake Ontario sends out a distress signal. The coordinates of the boat in trouble were $(49,64) .$ One rescue boat is at the coordinates (60,82) and a second Coast Guard craft is at coordinates $(58,47) .$ Assuming both rescue craft travel at the same rate, which one would get to the distressed boat the fastest?

Julie Silva

Numerade Educator

Problem 63

A man on the top of a building wants to have a guy wire extend to a point on the ground $20 \mathrm{ft}$ from the building. To the nearest foot, how long will the wire have to be if the building is 50 ft tall?

Zach Steedman

Numerade Educator

Problem 64

If we rent a truck and pay a $\$ 75 /$ day fee plus $\$ .20$ for every mile we travel, write a linear equation that would express the total cost $y,$ using $x$ to represent the number of miles we travel. Graph this function on your graphing calculator and find the total cost for one day if we travel $70 \mathrm{mi}$

Rishi Kavikondala

Rishi Kavikondala

Numerade Educator