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College Algebra 11th

Michael Sullivan

Chapter 2

Graphs

Educators


Problem 1

On the real number line, the origin is assigned the number_______________.

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Problem 2

If -3 and 5 are the coordinates of two points on the real number line, the distance between these points is ____________.

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Problem 3

If 3 and 4 are the legs of a right triangle, the hypotenuse is ________________.

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Problem 4

Use the converse of the Pythagorean Theorem to show that a triangle whose sides are of lengths $11,60,$ and 61 is a right triangle.

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Problem 5

The area $A$ of a triangle whose base is $b$ and whose altitude is $h$ is $A=$ ___________________.

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Problem 6

True or False Two triangles are congruent if two angles and the included side of one equals two angles and the included side of the other.

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Problem 7

If $(x, y)$ are the coordinates of a point $P$ in the $x y$ -plane, then $x$ is called the is the ______________________ of $P$, and $y$ is the _______________________ of $P$.

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Problem 8

The coordinate axes partition the $x y$ -plane into four sections called _______________.

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Problem 9

If three distinct points $P, Q,$ and $R$ all lie on a line, and if $d(P, Q)=d(Q, R),$ then $Q$ is called the _______________ of the line segment from $P$ to $R$.

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Problem 10

True or False. The distance between two points is sometimes a negative number.

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Problem 11

True or False The point ( - 1 , 4 ) lies in quadrant IV of the Cartesian plane.

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Problem 12

True or False. The midpoint of a line segment is found byaveraging the $x$ -coordinates and averaging the $y$ -coordinates of the endpoints.

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Problem 13

Multiple Choice. Which of the following statements is true for a point $(x, y)$ that lies in quadrant III?
(a) Both $x$ and $y$ are positive.
(b) Both $x$ and $y$ are negative.
(c) $x$ is positive, and $y$ is negative.
(d) $x$ is negative, and $y$ is positive.

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Problem 14

Multiple Choice Choose the expression that equals the
distance between two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$
(a) $\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$
(b) $\sqrt{\left(x_{2}+x_{1}\right)^{2}-\left(y_{2}+y_{1}\right)^{2}}$
(c) $\sqrt{\left(x_{2}-x_{1}\right)^{2}-\left(y_{2}-y_{1}\right)^{2}}$
(d) $\sqrt{\left(x_{2}+x_{1}\right)^{2}+\left(y_{2}+y_{1}\right)^{2}}$

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Problem 15

Plot each point in the xy-plane. State which quadrant or on what coordinate axis each point lies.
(a) $A=(-3,2)$
(b) $B=(6,0)$
(c) $C=(-2,-2)$
(d) $D=(6,5)$
(e) $E=(0,-3)$
(f) $F=(6,-3)$

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Problem 16

Plot each point in the xy-plane. State which quadrant or on what coordinate axis each point lies.
(a) $A=(1,4)$
(d) $D=(4,1)$
(b) $B=(-3,-4)$
(e) $E=(0,1)$
(c) $C=(-3,4)$
(f) $F=(-3,0)$

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Problem 17

Plot the points $(2,0),(2,-3),(2,4),(2,1),$ and $(2,-1) .$ Describe the set of all points of the form $(2, y),$ where $y$ is a real number.

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Problem 18

Plot the points $(0,3),(1,3),(-2,3),(5,3),$ and $(-4,3) .$ Describe the set of all points of the form $(x, 3),$ where $x$ is a real number.

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Problem 19

Find the distance $d$ between the points $P_{1}$ and $P_{2}$.

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Problem 20

Find the distance $d$ between the points $P_{1}$ and $P_{2}$.

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Problem 21

Find the distance $d$ between the points $P_{1}$ and $P_{2}$.

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Problem 22

Find the distance $d$ between the points $P_{1}$ and $P_{2}$.

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Problem 23

Find the distance $d$ between the points $P_{1}$ and $P_{2}$.
$$P_{1}=(3,-4) ; \quad P_{2}=(5,4)$$

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Problem 24

Find the distance $d$ between the points $P_{1}$ and $P_{2}$.
$$P_{1}=(-1,0) ; \quad P_{2}=(2,4)$$

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Problem 25

Find the distance $d$ between the points $P_{1}$ and $P_{2}$.
$$P_{1}=(-7,3) ; \quad P_{2}=(4,0)$$

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Problem 26

Find the distance $d$ between the points $P_{1}$ and $P_{2}$.
$$P_{1}=(2,-3) ; \quad P_{2}=(4,2)$$

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Problem 27

Find the distance $d$ between the points $P_{1}$ and $P_{2}$.
$$P_{1}=(5,-2) ; \quad P_{2}=(6,1)$$

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Problem 28

Find the distance $d$ between the points $P_{1}$ and $P_{2}$.
$$P_{1}=(-4,-3) ; \quad P_{2}=(6,2)$$

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Problem 29

Find the distance $d$ between the points $P_{1}$ and $P_{2}$.
$$P_{1}=(-0.2,0.3) ; \quad P_{2}=(2.3,1.1)$$

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Problem 30

Find the distance $d$ between the points $P_{1}$ and $P_{2}$.
$$P_{1}=(1.2,2.3) ; \quad P_{2}=(-0.3,1.1)$$

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Problem 31

Find the distance $d$ between the points $P_{1}$ and $P_{2}$.
$$P_{1}=(a, b) ; \quad P_{2}=(0,0)$$

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Problem 32

Find the distance $d$ between the points $P_{1}$ and $P_{2}$.
$$P_{1}=(a, a) ; \quad P_{2}=(0,0)$$

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Problem 33

Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.
$$A=(-2,5) ; \quad B=(1,3) ; \quad C=(-1,0)$$

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Problem 34

Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.
$$A=(-2,5) ; \quad B=(12,3) ; \quad C=(10,-11)$$

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Problem 35

Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.
$$A=(-5,3) ; \quad B=(6,0) ; \quad C=(5,5)$$

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Problem 36

Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.
$$A=(-6,3) ; \quad B=(3,-5) ; \quad C=(-1,5)$$

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Problem 37

Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.
$A=(4,-3) ; \quad B=(0,-3) ; \quad C=(4,2)$

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Problem 38

Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.
$A=(4,-3) ; \quad B=(4,1) ; \quad C=(2,1)$

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Problem 39

Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.
$P_{1}=(3,-4) ; \quad P_{2}=(5,4)$

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Problem 40

Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.
$P_{1}=(-2,0) ; \quad P_{2}=(2,4)$

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Problem 41

Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.
$P_{1}=(-1,4) ; \quad P_{2}=(8,0)$

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Problem 42

Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.
$P_{1}=(2,-3) ; \quad P_{2}=(4,2)$

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Problem 43

Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.
$P_{1}=(7,-5) ; \quad P_{2}=(9,1)$

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Problem 44

Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.
$P_{1}=(-4,-3) ; \quad P_{2}=(2,2)$

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Problem 45

Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.
$P_{1}=(a, b) ; \quad P_{2}=(0,0)$

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Problem 46

Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.
$P_{1}=(a, a) ; \quad P_{2}=(0,0)$

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Problem 47

If the point (2,5) is shifted 3 units to the right and 2 units down, what are its new coordinates?

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Problem 48

If the point (-1,6) is shifted 2 units to the left and 4 units up, what are its new coordinates?

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Problem 49

Find all points having an $x$ -coordinate of 3 whose distance from the point (-2,-1) is $13 .$
(a) By using the Pythagorean Theorem.
(b) By using the distance formula.

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Problem 50

Find all points having a $y$ -coordinate of -6 whose distance from the point (1,2) is $17 .$
(a) By using the Pythagorean Theorem.
(b) By using the distance formula.

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Problem 51

Find all points on the $x$ -axis that are 6 units from the point (4,-3)

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Problem 51

Find all points on the $x$ -axis that are 6 units from the point (4,-3)

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Problem 52

Find all points on the $y$ -axis that are 6 units from the point (4,-3)

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Problem 53

Suppose that $A=(2,5)$ are the coordinates of a point in the $x y$ -plane.
(a) Find the coordinates of the point if $A$ is shifted 3 units to the left and 4 units down.
(b) Find the coordinates of the point if $A$ is shifted 2 units to the left and 8 units up.

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Problem 54

Plot the points $A=(-1,8)$ and $M=(2,3)$ in the $x y$ -plane. If $M$ is the midpoint of a line segment $A B,$ find the coordinates of $B$.

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Problem 55

The midpoint of the line segment from $P_{1}$ to $P_{2}$ is (-1,4) . If $P_{1}=(-3,6),$ what is $P_{2} ?$

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Problem 56

The midpoint of the line segment from $P_{1}$ to $P_{2}$ is (5,-4) . If $P_{2}=(7,-2),$ what is $P_{1} ?$

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Problem 57

The medians of a triangle are the line segments from each vertex to the midpoint of the opposite side (see the figure). Find the lengths of the medians of the triangle with vertices at $A=(0,0), B=(6,0),$ and $C=(4,4)$

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Problem 58

An equilateral triangle has three sides of equal length. If two vertices of an equilateral triangle are (0,4) and (0,0) find the third vertex. How many of these triangles are possible?

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Problem 59

Find the length of each side of the triangle determined by the three points $P_{1}, P_{2},$ and $P_{3}$. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or
both. (An isosceles triangle is one in which at least two of the sides are of equal length. )
$P_{1}=(2,1) ; \quad P_{2}=(-4,1) ; \quad P_{3}=(-4,-3)$

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Problem 60

Find the length of each side of the triangle determined by the three points $P_{1}, P_{2},$ and $P_{3}$. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or
both. (An isosceles triangle is one in which at least two of the sides are of equal length. )
$P_{1}=(-1,4) ; \quad P_{2}=(6,2) ; \quad P_{3}=(4,-5)$

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Problem 61

Find the length of each side of the triangle determined by the three points $P_{1}, P_{2},$ and $P_{3}$. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or
both. (An isosceles triangle is one in which at least two of the sides are of equal length. )
$P_{1}=(-2,-1) ; \quad P_{2}=(0,7) ; \quad P_{3}=(3,2)$

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Problem 62

Find the length of each side of the triangle determined by the three points $P_{1}, P_{2},$ and $P_{3}$. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or
both. (An isosceles triangle is one in which at least two of the sides are of equal length. )
$P_{1}=(7,2) ; \quad P_{2}=(-4,0) ; \quad P_{3}=(4,6)$

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Problem 63

A major league baseball "diamond" is actually a square 90 feet on a side (see the figure). What is the distance directly from home plate to second base (the diagonal of the square $) ?$

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Problem 64

The layout of a Little League playing field is a square 60 feet on a side. How far is it directly from home plate to second base (the diagonal of the square)?

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Problem 65

Refer to Problem $63 .$ Overlay a rectangular coordinate system on a major league baseball diamond so that the origin is at home plate, the positive $x$ -axis lies in the direction from home plate to first base, and the positive $y$ -axis lies in the direction from home plate to third base.
(a) What are the coordinates of first base, second base, and third base? Use feet as the unit of measurement.
(b) If the right fielder is located at (310,15) how far is it from the right fielder to second base?
(c) If the center fielder is located at $(300,300),$ how far is it from the center fielder to third base?

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Problem 66

Refer to Problem 64. Overlay $\begin{array}{llllll}\text { a rectangular } & \text { coordinate } & \text { system } & \text { on } & \text { a } & \text { Little } & \text { League }\end{array}$ baseball diamond so that the origin is at home plate, the positive $x$ -axis lies in the direction from home plate to first base, and the positive $y$ -axis lies in the direction from home plate to third base.
(a) What are the coordinates of first base, second base, and third base? Use feet as the unit of measurement.
(b) If the right fielder is located at $(180,20),$ how far is it from the right fielder to second base?
(c) If the center fielder is located at $(220,220),$ how far is it from the center fielder to third base?

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Problem 67

A Ford Focus and a Freightliner Cascadia truck leave an intersection at the same time. The Focus heads east at an average speed of 60 miles per hour, while the Cascadia heads south at an average speed of 45 miles per hour. Find an expression for their distance apart $d$ (in miles) at the end of $t$ hours.

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Problem 68

A hot-air balloon, headed due east at an average speed of 15 miles per hour and at a constant altitude of 100 feet, passes over an intersection (see the figure). Find an expression for the distance $d$ (measured in feet) from the balloon to the intersection $t$ seconds later.

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Problem 69

When a draftsman draws three lines that are to intersect at one point, the lines may not intersect as intended and subsequently will form an error triangle. If this error triangle is long and thin, one estimate for the location of the desired point is the midpoint of the shortest side. The figure shows one such error triangle.
(a) Find an estimate for the desired intersection point.
(b) Find the distance from (1.4,1.3) to the midpoint found in part (a).

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Problem 70

The figure illustrates the net sales growth of costco Wholesale Corporation from 2013 through 2017 . Use the midpoint formula to estimate the net sales of costco Wholesale Corporation in 2015. How does your result compare to the reported value of $\$ 113.67$ billion? Source: Costco Wholesale Corporation 2017 Annual Report

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Problem 71

Poverty Threshold Poverty thresholds are determined by the U.S. Census Bureau. A poverty threshold represents the minimum annual household income for a family not to be considered poor. In $2009,$ the poverty threshold for a family of four with two children under the age of 18 years was $ 21,756 $ . In 2017, the poverty threshold for a family of four with two children under the age of 18 years was 24,858.
Assuming that poverty thresholds increase in a straight-line fashion, use the midpoint formula to estimate the poverty threshold for a family of four with two children under the age of 18 in $2013 .$ How does your result compare to the actual poverty threshold in 2013 of $\$ 23,624 ?$
Source: U.S. Census Bureau

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Problem 72

Verify that the points $(0,0),(a, 0),$ and $\left(\frac{a}{2}, \frac{\sqrt{3} a}{2}\right)$ are the vertices of an equilateral triangle.Then show that the midpoints of the three sides are the vertices of a second equilateral triangle.

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Problem 73

Find the midpoint of each diagonal of a square with side of length $s$. Draw the conclusion that the diagonals of a square intersect at their midpoints. [Hint: Use $(0,0),(0, s),(s, 0),$ and $(s, s)$ as the vertices of the square.]

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Problem 74

A point $P$ is equidistant from (-5,1) and $(4,-4) .$ Find the coordinates of $P$ if its $y$ -coordinate is twice its $x$ -coordinate.

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Problem 75

For any parallelogram, prove that the sum of the squares of the lengths of the sides equals the sum of the squares of the lengths of the diagonals.
[Hint: Use $(0,0),(a, 0),(a+b, c),$ and $(b, c)$ as the vertices of the parallelogram.

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Problem 76

Write a paragraph that describes a Cartesian plane. Then write a second paragraph that describes how to plot points in the Cartesian plane. Your paragraphs should include the terms "coordinate axes," "ordered pair," "coordinates," "plot," " $x$ -coordinate," and " $y$ -coordinate."

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Problem 77

Determine the domain of the variable $x$ in the expression:
$$
\frac{3 x+1}{2 x-5}
$$

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Problem 78

Find the value of $\frac{x^{2}-3 x y+2}{5 x-2 y}$ if $x=4$ and $y=7$

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Problem 79

Express $(5 x-2)(3 x+7)$ as a polynomial in standard form.

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Problem 80

Reduce the rational expression to lowest terms:
$$
\frac{3 x^{2}+7 x+2}{3 x^{2}-11 x-4}
$$

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Problem 81

Factor the expression completely:
$$
6(x+1)^{3}(2 x-5)^{7}-5(x+1)^{4}(2 x-5)^{6}
$$

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Problem 82

Find the real solution $(s),$ if any, of each equation.
$3 x^{2}-7 x-20=0$

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Problem 83

Find the real solution $(s),$ if any, of each equation.
$\frac{x}{x+3}+\frac{1}{x-3}=1$

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Problem 84

Find the real solution $(s),$ if any, of each equation.
$|7 x-4|=31$

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Problem 85

Solve the inequality $5(x-3)+2 x \geq 6(2 x-3)-7$ Express the solution using interval notation. Graph the solution set.

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Problem 86

Multiply $(7+3 i)(1-2 i)$. Write the answer in the form $a+b i$.

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