Introduction to Algebra

• Monomial: an algebraic expression that contains only one term.

• Binomial: an algebraic expression that contains two terms.

• Trinomial: an algebraic expression that contains three terms.

Combining Terms

• Use the distributive property: a (b + c) = ab + ac
• Multiply coefficients and add exponents with the same base. Apply a^m aⁿ = a^m+n

• Split the polynomial into several parts.
• Divide a monomial by a monomial. Apply $\frac{a^m}{a^n}=a^{m-n}$.

a. 5x³ + 8x² + 2x                                                                                             

b. –$\frac{2}{3}$y⁴ + 9a² + a – 1                                                                                      

a. 7x + 10y – 8x + 9y

b. 12a² – 33b + 2b – 6a²

c. 13n + 5(6n – m²) + 7(2m² + 3n)

a. 15a² + 9 – (5a² – 4)

b. (-13x + 9y) – 6(x – 5y)                               

c. 5(ab – 2xy) – 6(-2ab + 3xy)

d. (5y – 7) (8y + 9)

e. (7r – 2t) (3r + 4t²)

f. (x – $\frac{1}{3}$) (x – $\frac{2}{3}$)

Equations

• Add, subtract, multiply or divide the same quantity to both sides of an equation to obtain a valid equation.
• Remember to always do the same thing to both sides of the equation (balance).

• When trying to figure out the correct operation ( + , – , × , ÷ , etc.) in a word problem it is important to pay attention to keywords (clues to what the problem is asking).
• Identifying keywords and pulling out relevant information that appear in the word problem are effective ways for solving mathematical word problems.

a. x – 7 = 12

b. y + $\frac{3}{8}$ = $\frac{5}{8}$

c. $\frac{3x}{2}$ = $\frac{9}{16}$

d. 14t + 5 = 8

e. 7(x – 3) + 3x – 5 = 2(5 – 4x)

f. $\frac{1}{7}$(y + 12) = 4y – $\frac{3}{7}$y

g. 0.5t + 0.05 = 0.025

h. $\frac{x}{4}+\frac{2}{5}=-\frac{x}{2}-\frac{1}{5}$

a. Susan has \$375 in her checking account. If she makes a deposit of y dollars, how much in total will she have in her account?

b. Mark weighs 175 pounds. If he loses y pounds, how much will he weigh?

c. A piece of wire 45 metres long was cut in two pieces and one piece is w metres long. How long is the other piece?

d. Emily made 4 dozen muffins. If it cost her x dollars, what was her cost per dozen muffins? What was her cost per muffin?

Measurement Systems

• Identify the number of places to move the decimal point.

- Convert a smaller unit to a larger unit: move the decimal point to the left.

- Convert a larger unit to a smaller unit:  move the decimal point to the right.

Example 1.3.1

326 mm = (?) m

• Identify mm (millimeters) and m (meters) on the conversion table.

Count places from mm to m:                      3 places left

• Move 3 decimal places.          (1 m = 1000 mm)

Convert a smaller unit (mm) to a larger (m) unit: move the decimal point to the left.

326. mm = 0.326 m

Move the decimal point three places to the left (326 = 326.).

Example 1.3.2

4.675 hg = (?) g

• Identify hg (hectograms) and g (grams) on the conversion table.

Count places from hg to g:                      2 places right

• Move 2 decimal places.        (1 hg = 100 g)

Convert a larger unit (hg) to a smaller (g) unit:  move the decimal point to the right.

4.765 hg = 476.5 g

Move the decimal point two places to the right.

• Write the original term as a fraction (over 1).

• Write the conversion formula as a fraction $\frac{1}{(\;)}$ or $\frac{(\;)}{1}$.

• Put the desired or unknown unit on the top.
• Multiply the original term by $\frac{1}{(\;)}$ or $\frac{(\;)}{1}$. (Cancel out the same units).

• A cube takes up 1 cm³ of space (1 cm × 1 cm × 1 cm = 1cm³).
• A cube holds 1 mL of water and has a mass of 1 gram at 4°C.
• 1 cm³ = 1 mL = 1 g

a. 439 mm = ( ? ) m

b. 48.3 mL = ( ? ) kL

c. 7230 g = ( ? ) kg

d. 52 cm = ( ? ) mm

a. 7 m – 3000 mm = ( ? ) mm

b. 63 kg + 6 g = ( ? ) g

a. 38 cm³ = (       ) g

b. 5 L = (       ) cm³

c. 18 L of water has a volume of ___________ cm³ .

d. A water tank measures 45 cm by 35 cm by 25 cm. How many kilolitres of water will it hold?

The ratio of a and b is:     a to b      or      a : b      or      $\frac{a}{b}$

• Write a ratio in lowest terms (simplify):

- Write the ratio in a fractional form.

- Simplify and drop the units if given (as they cancel each other out).

• Write a rate in lowest terms (simplify the rate):

• Some unit rates:

- Miles (or kilometres) per hour (or minute).

- Cost (dollars/cents) per item or quantity.

- Earnings (dollars) per hour (or week).

• Cross multiply: multiply along two diagonals.     $\frac{a}{b}=\frac{c}{d}$
• Solve for the unknown.

• Percent proportion method
• Translation (translate the words into mathematical symbols.)

• What ____  x :    the word “what” represents an unknown quantity x.
• Is      ____  = :    the word “is” represents an equal sign.
• of     ____  × :    the word “of” represents a multiplication sign.
• %    to     decimal:    always change the percent to a decimal.

a. 4 nickels to 16 nickels.

b. 350 people for 1500 tickets.

c. 160 kilometres per 740 minutes.

a. What % of students were unemployed in 2015?

b. If the % of unemployed students remained the same in 2016, find the number of unemployed students in 2016.

c. If the % of unemployed students increased by an additional 5% in 2017, find the total number of students in 2017.

Exponents

• Product rule: when multiplying two powers with the same base, keep the base and add the exponents.

a^m aⁿ = a^{m + n}            aⁿ   or   Base^Exponent

• Quotient rule: when dividing two powers with the same base, keep the base and subtract the exponents.

$\frac{a^m}{a^n}=a^{m-n}$

• Power rule: when raising an expression to a power, we multiply each exponent inside the parentheses by the power outside the parentheses.

(a^m)ⁿ = a^mn,        (a^m · bⁿ)^p = a^mp b^np,        $(\frac{a^m}{b^n})^p=\frac{a^{mp}}{b^{np}}$

• Negative exponent: a negative exponent is the reciprocal of the number with a positive exponent.

$a^{-n}=\frac{1}{a^n}$,  $\frac{1}{a^{-n}}=a^n$              a⁻ⁿ is the reciprocal of aⁿ.

• Fractional exponent: a fractional exponent is a different way of writing a radical (i.e. root) sign. The base is first taken to the exponent of m, then the nth root is found to obtain the power.

$a^{\frac{m}{n}} = {\sqrt[n]{a}^{m}} = {\sqrt[n]{a^{m}}}$

 Simplifying Exponential Expressions

• Remove parentheses using “power rule” if necessary.             (a^m bⁿ)^p = a^mp b^np
• Regroup coefficients and variables.
• Use “product rule” and “quotient rule”.                                   a^m aⁿ = a^{m + n} , $\frac{a^m}{a^n}=a^{m-n}$
• Simplify.
• Use the “negative exponent” rule to make all exponents positive if necessary.

a. 4x² + 5y,    for x = 1,    y = 4

b. (2a)³ – 3b,    for a = 5,    b = 6

a. (-92)¹

b. y⁴ y³

c. $\frac{x^9}{x^6}$

d. 13a^-1

e. (3a² · b³)⁴

f. $\frac{5x^5y^{-6}}{11^0x^3y^4}$

g. $(\frac{u^{-2}v^3}{w^{-4}})^{-3}$       

a. 45,600,000

b. 0.00000523

a. 3.578 × 10³

b. 4.3 × 10^-5

Graphing a Linear Equation

Example 2.1.1

Graph the line: y = 3x + 2 Solution We need to find the coordinates of at least two points. We arbitrarily choose x = -1 , x = 0 , and x = 1. If x = -1, then y = 3(-1) + 2 or -1. Therefore, (-1, -1) is a point on this line. If x = 0, then y = 3(0) + 2 or y = 2. Hence the point (0, 2). If x = 1, then y = 5, and we get the point (1, 5). Below, the results are summarized, and the line is graphed.

X	-1	0	1
Y	-1	2	5

Example 2.1.2

Graph the line: 2x + y = 4

Solution

Again, we need to find coordinates of at least two points.

We arbitrarily choose x = -1, x = 0 and y = 2.

If x = -1, then 2(-1) + y = 4 which results in y = 6. Therefore, (-1, 6) is a point on this line.

If x = 0, then 2(0) + y = 4, which results in y = 4. Hence the point (0, 4).

If y = 2, then 2x + 2 = 4, which yields x = 1, and gives the point (1, 2). The table below shows the points, and the line is graphed.

x	-1	0	1
y	6	4	2

Example 2.1.3

Find the intercepts of the line: 2x − 3y = 6, and graph.

Solution

To find the x-intercept, we let y = 0 in our equation, and solve for x.

$2x - 3(0) = 6$

$2x - 0 = 6$

$2x = 6$

$x = 3$

Therefore, the x-intercept is 3.

Similarly by letting x = 0, we obtain the y-intercept which is -2.

Note: If the x-intercept is 3, and the y-intercept is -2, then the corresponding points are (3, 0) and (0, -2), respectively.

Example 2.1.4

Graph the line given by the parametric equations: x = 3 + 2t , y = 1 + t

Solution

Let t = 0, 1 and 2, and then for each value of t find the corresponding values for x and y.

The results are given in the table below.

t	0	1	2
x	3	5	7
y	1	2	3

Horizontal and Vertical Lines

Example 2.1.5

Graph the lines: x = -2 , and y = 3.

Solution

The graph of the line x = -2 is a vertical line that has the x-coordinate -2 no matter what the y-coordinate is. Therefore, the graph is a vertical line passing through (-2, 0).

The graph of the line y = 3, is a horizontal line that has the y-coordinate 3 regardless of what the x-coordinate is. Therefore, the graph is a horizontal line that passes through (0, 3).

Note: Most students feel that the coordinates of points must always be integers. This is not true, and in real life situations, not always possible. Do not be intimidated if your points include numbers that are fractions or decimals.

Practice questions

a. (2, ?) (? , 6)

b. (0, ?) (?, 0)

1. Find the slope of a line if two points are given.
2. Graph the line if a point and the slope are given.
3. Find the slope of the line that is written in the form y = mx + b.
4. Find the slope of the line that is written in the form Ax+ By = c.

$\frac{rise}{run},\frac{vertical\;change}{horizontal\;change},\frac{\Delta y}{\Delta x}etc.$

Example 2.2.1

Find the slope of the line that passes through the points (-2, 3) and (4, -1), and graph the line.

Solution

Let (x₁, y₁) = (-2, 3) and (x₂, y₂) = (4, −1) then the slope

$m=\frac{-1-3}{4-(-2)}=-\frac{4}{6}=-\frac{2}{3}$

$m=\frac{-3-(-1)}{-2-4}=\frac{4}{-6}=-\frac{2}{3}$

Example 2.2.2

Find the slope of the line that passes through the points (2, 3) and (2, -1), and graph.

Solution

Let (x1, y1) = (2, 3) and (x2, y2) = (2, -1) then the slope

$m=\frac{-1-3}{2-2}=-\frac{4}{0}=$ undefined

Note: The slope of a vertical line is undefined.

Example 2.2.3

Graph the line that passes through the point (1, 2) and has slope $-\frac{3}{4}$.

Solution

Slope equals $\frac{rise}{run}$. The fact that the slope is $\frac{-3}{4}$ , means that for every rise of -3 units (fall of 3 units) there is a run of 4. So if from the given point (1, 2) we go down 3 units and go right 4 units, we reach the point (5, -1). The following graph is obtained by connecting these two points.

Alternatively, since $\frac{3}{-4}$ represents the same number, the line can be drawn by starting at the point (1, 2) and choosing a rise of 3 units followed by a run of -4 units. So from the point (1, 2), we go up 3 units, and to the left 4, thus reaching the point (-3, 5) which is also on the same line. See the figure below.

Example 2.2.4

Find the slope of the line 2x + 3y = 6.

Solution

In order to find the slope of this line, we will choose any two points on this line.

Again, the selection of x and y intercepts seems to be a good choice. The x-intercept is (3, 0), and the y-intercept is (0, 2). Therefore, the slope is

$m=\frac{2-0}{0-3}=-\frac{2}{3}$

The graph below shows the line and the intercepts: x and y.

Practice questions

a. (2, 3) and (5, 9)

b. (6, -5) and (4, -1)

c. (-3, -5) and (-1, -7)

a. $y = -2x + 1$

b. $3x - 4y = 12$

c. $2x - y = 6$

a. (1, 2) and m = -3/4

b. (0, 2) and m = -2

1. Find an equation of a line if a point and the slope are given.
2. Find an equation of a line if two points are given.

Practice questions

a. Passes through (3, 5) and (2, -1).

b. Passes through (5, -2) and m = $2/5$.

c. Passes through (2, -5) and its x-intercept is 4.

d. Passes through (-3, -4), and (-5, 2).

e. Is a horizontal line passing through (2, -1).

f. Has an x-intercept = 3 and y-intercept = 4.

a. Passes through (-4, -2) and m = $3/4$

b. Passes through (2, -3) and (5, 1).

More Applications

1. Solve a linear system in two variables.
2. Find the equilibrium point when a demand and a supply equation are given.
3. Find the break-even point when the revenue and the cost functions are given.

$3x-1=-x+7$

$4x = 8$

$x = 2$

Supply, Demand and the Equilibrium Market Price

Example 2.4.10

The supply curve for a product is y = 1.5x + 10 and the demand curve for the same product is y = -2.5x + 34, where x is the price and y the number of items produced. Find the following:

a. How many items will be supplied at a price of \$10?

b. How many items will be demanded at a price of \$10?

c. Determine the equilibrium price.

d. How many items will be produced at the equilibrium price?

Solution

a. We substitute x = 10 in the supply equation, y = 1.5x + 10, and the answer is y = 25.

b. We substitute x = 10 in the demand equation, y = -2.5x + 34, and the answer is y = 9.

c. By letting the supply equal the demand, we get:

$1.5x + 10 =-2.5x + 34$

$4x = 24$

$x = 6$

d. We substitute x = 6 in either the supply or the demand equation and we get y = 19. The graph below shows the intersection of the supply and the demand functions and their point of intersection, (6, 19).

Break-Even Point

Example 2.4.11

If the revenue function of a product is R = 5x and the cost function is y = 3x + 12, find the following:

a. If 4 items are produced, what will the revenue be?

b. What is the cost of producing 4 items?

c. How many items should be produced to break-even?

d. What will be the revenue and the cost at the break-even point?

Solution

a. We substitute x = 4 in the revenue equation R = 5x, and the answer is R = 20.

b. We substitute x = 4 in the cost equation C = 3x + 12, and the answer is C = 24.

c. By letting the revenue equal the cost, we get:

$5x = 3x+12$

$x=6$

d. We substitute x = 6 in either the revenue or the cost equation, and we get R = C = 30. The graph below shows the intersection of the revenue and the cost functions and their point of intersection, (6, 30).

Practice questions

Function Notation

Tables as Functions

Solving and Evaluating Functions

Graphs as Functions

Example 3.1.8

Which of these graphs defines a function y = f(x)?

Looking at the three graphs above, the first two define a function y = f(x), since for each input value along the horizontal axis there is exactly one output value corresponding, determined by the y-value of the graph. The 3^rd graph does not define a function y = f(x) since some input values, such as x = 2, correspond with more than one output value.

Example 3.1.9

Given the graph shown:

a) Evaluate f(2) b) Solve f(x) = 4   Solution a) To evaluate f(2), we find the input of x = 2 on the horizontal axis. Moving up to the graph gives the point (2, 1), giving an output of y = 1. So f(2) = 1. b) To solve f(x) = 4, we find the value 4 on the vertical axis because if f(x) = 4 then 4 is the output. Moving horizontally across the graph gives two points with the output of 4: (-1,4) and (3,4). These give the two solutions to f(x) = 4: x = -1 or x = 3 This means f(-1) = 4 and f(3) = 4, or when the input is -1 or 3, the output is 4.

Formulas as Functions

Practice questions

Notation

Example 3.2.3

Describe the intervals of values shown on the line graph below using set builder and interval notations.

To describe the values, x , that lie in the intervals shown above we would say, “x is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.” As an inequality it is: 1 ≤ x ≤ 3 or x > 5. In set builder notation: {x | 1 ≤ x ≤ 3 or x > 5}. In interval notation: [1, 3] ∪ (5, ∞).

Domain and Range from Graphs

Example 3.2.4

Determine the domain and range of the graph below:

In the graph above, the input quantity along the horizontal axis appears to be “year”, which we could notate with the variable y. The output is “thousands of barrels of oil per day”, which we might notate with the variable b, for barrels. The graph would likely continue to the left and right beyond what is shown, but based on the portion of the graph that is shown to us, we can determine the domain is 1975 ≤ y ≤ 2008 , and the range is approximately 180 ≤ b ≤ 2010 . In interval notation, the domain would be [1975, 2008] and the range would be about [180, 2010]. For the range, we have to approximate the smallest and largest outputs since they don’t fall exactly on the grid lines.

Domain and Range from Formulas

• The square root of negative values is non-real.
• We cannot divide by zero.

Piecewise Functions

Example 3.2.8

Sketch a graph of the function $f(x)=\begin{cases} 0\quad if \quad x\le1\\ x-1\quad if \quad 1<x\le4\\ 2\quad if \quad x>4\\ \end{cases}$ We can imagine graphing each function, then limiting the graph to the indicated domain. At the endpoints of the domain, we put open circles to indicate where the endpoint is not included, due to a strictly-less-than inequality, and a closed circle where the endpoint is included, due to a less-than-or-equal-to inequality. The first and last parts are constant functions, where the output is the same for all inputs. The middle part we might recognize as a line, and could graph by evaluating the function at a couple inputs and connecting the points with a line.

Now that we have each piece individually, we combine them onto the same graph. When the first and second parts meet at x = 1, we can imagine the closed dot filling in the open dot. Since there is no break in the graph, there is no need to show the dot.

Practice questions

a. Domain

b. Range

• A population of rats increases by 40 rats per week
• A barista earns \$9 per hour (dollars per hour)
• A farmer plants 60,000 onions per acre
• A car can drive 27 miles per gallon
• A population of grey whales decreases by 8 whales per year
• The amount of money in your university account decreases by \$4,000 per quarter

Example 3.3.2

Given the function g(t) shown here, find the average rate of change on the interval [0, 3].

At t = 0, the graph shows g(0) = 1 At t = 3, the graph shows g(3) = 4 The output has changed by 3 while the input has changed by 3, giving an average rate of change of: $\frac{4-1}{3-0}=\frac{3}{3}=1$

Graphical Behaviour of Functions

Example 3.3.6

Given the function p(t) graphed here, on what intervals does the function appear to be increasing?

The function appears to be increasing from t = 1 to t = 3, and from t = 4 on. In interval notation, we would say the function appears to be increasing on the interval (1, 3) and the interval (4, ∞).

Practice questions

 

Quadratic Functions

Example 3.4.1

A backyard farmer wants to enclose a rectangular space for a new garden. She has purchased 80 feet of wire fencing to enclose 3 sides, and will put the 4^th side against the backyard fence. Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length L.

In a scenario like this involving geometry, it is often helpful to draw a picture. It might also be helpful to introduce a temporary variable, W, to represent the side of fencing parallel to the 4^th side or backyard fence. Since we know we only have 80 feet of fence available, we know that L + W + L = 80 , or more simply, 2L + W = 80 . This allows us to represent the width, W, in terms of L:    W = 80 – 2L Now we are ready to write an equation for the area the fence encloses. We know the area of a rectangle is length multiplied by width, so: A = LW = L(80 – 2L) A(L) = 80L – 2L² This formula represents the area of the fence in terms of the variable length L.

Short Run Behaviour: Vertex

Example 3.4.2

Sketch a graph of $g(x) = \frac{1}{2}(x+2)^2-3$:

We can create a table of values, which we can use to plot several points and connect them with a smooth curve.

x	g(x)
-5	1.5
-4	-1
-3	-2.5
-2	-3
-1	-2.5
0	-1
1	1.5

Forms of Quadratic Functions: The standard form of a quadratic function is $f(x)=ax^2+bx+c$. The vertex form of a quadratic function is $f(x)=a(x-h)^2+k$. The vertex of the quadratic function is located at (h, k), where h and k are the numbers in the vertex form of the function. When a > 0, the graph of the quadratic will open upwards. When a < 0, the graph of the quadratic will open downwards.

a > 0

a < 0

$h=-\frac{b}{2a}$,   $k=f(h)=f(\frac{-b}{2a})$

Short run Behavior: Intercepts

zero horizontal intercepts

one horizontal intercept

two horizontal intercepts

Practice questions

Exponential Functions

2	200	225
4	300	506
6	400	1139
8	500	2563
10	600	5767

Finding Equations of Exponential Functions

Graphs of Exponentials

1. This function is positive for all values of x.
2. As x increases, the function grows faster and faster (the rate of change increases).
3. As x decreases, the function values grow smaller, approaching zero.
4. This is an example of exponential growth.

$f(-x)=2^{-x}=(2^{-1})^x=(\frac{1}{2})^x=g(x)$

Example 4.1.6

Sketch a graph of $f(x)=4(\frac{1}{3})^x$ This graph will have a vertical intercept at (0,4), and pass through the point (1, $\frac{4}{3}$) . Since b < 1, the graph will be decreasing towards zero. Since a > 0, the graph will be concave up. We can also see from the graph the long run behavior: as x→∞ , f(x)→0 and as x→-∞ , f(x)→∞ .

Example 4.1.7

Match each equation with its graph.

$f(x)=2(1.3)^x$ $g(x)=2(1.8)^x$ $h(x)=4(1.3)^x$ $k(x)=4(0.7)^x$

The graph of k(x) is the easiest to identify, since it is the only equation with a growth factor less than one, which will produce a decreasing graph. The graph of h(x) can be identified as the only growing exponential function with a vertical intercept at (0, 4). The graphs of f(x) and g(x) both have a vertical intercept at (0, 2), but since g(x) has a larger growth factor, we can identify it as the graph increasing faster.

Compound Interest

A Limit to Compounding

Practice questions

C ={Ken, Bob, Tran, Shanti, Eric}

Venn Diagrams

Practice questions

a. $V\cup W$

b. $V^c\cap W$

a. $A\cap C$

b. $({A \cup B})^c\cap C$

a. The number of students subscribing to Amazon Prime but not the other two streaming services.

b. The number of students subscribing to Netflix or Amazon Prime but not Disney+.

c. The number of students not subscribing to any of these services.

Sample Spaces and Probability

HH HT TH TT

Practice questions

a. P (a king)

b. P (any suit other than hearts)

a. P (red)

b. P (red or blue)

a. P (the sum of the dice is 5)

b. P (the sum of the dice is 3 or 6)

a. P (they have two boys and two girls)

b. P (they have three or more girls)

Binomial Probability

1. If a coin is flipped 10 times, what is the probability that it will fall heads 3 times?
2. If a basketball player makes 3 out of every 4 free throws, what is the probability that he will make 7 out of 10 free throws in a game?
3. If a medicine cures 80% of the people who take it, what is the probability that among the 10 people who take the medicine, 6 will be cured?
4. If a microchip manufacturer claims that only 4% of their chips are defective, what is the probability that among the 60 chips chosen, exactly three are defective?
5. If a telemarketing executive has determined that 15% of the people contacted will purchase the product, what is the probability that among the 12 people who are contacted, 2 will buy the product?

Practice questions

a. Only one basket?

a. Three baskets?

c. At least three baskets?

a. 25 will be cured?

b. 26 will be cured?

c. 27 will be cured?

d. At least 25 will be cured?

a. One contaminated product?

b. Two contaminated products?

Markov Chains

ET = E

or,

$\left[\begin{array}{cc} 1/3 & 2/3\end{array}\right] \left[ \begin{array}{cc} 1/3 & 2/3 \\ 1/3 & 2/3 \end{array}\right]=\left[\begin{array}{cc} 1/3 & 2/3\end{array}\right]$

Practice questions

a. If the player shot on the first play, what is the probability that he will pass on the third play?

b. What is the long-term shoot vs. pass distribution of this player?

a. What is the percentage of people in the community that reported texting and driving after the campaign?

b. If the campaign were to be repeated multiple times, what is the long-range trend in terms of the lowest rate that texting and driving can be reduced to in this community?

a. Create a transition matrix for this scenario.

b. If the company employs 8500 workers, how many slip, trip and fall accidents were reported in year 2?

c. If the program continued for another year, how many accidents would be reported in year 3?

d. If the training program were to be repeated for many years, what is the lowest prevalence of slip, trip or fall accidents that could be achieved?

• Chapter 1: Algebra Review
• Chapter 2: Linear Equations and Graphs
• Chapter 3: Introduction to Functions
• Chapter 4: Exponential and Logarithmic Functions
• Chapter 5: Sets and Counting
• Chapter 6: Probability - Part 1
• Chapter 7: Probability - Part 2

Chapter 1 practice question answers