Chapter 2.1 – Open Sentences and Solution Sets
The difference between an expression and an equation:
Expression: does not contain an equals sign
Equation: a statement that 2 mathematical expressions are equal. Only statements with the symbols
=, ≠, <, >, ≤, ≥ are equations (math sentences).
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open sentences contain variables that represent unknown quantities and are neither true or
false.
Closed sentences contain no variables and are either true or false
When given a replacement set or domain set you must substitute every value in the replacement set to
determine which, if any, of the values in the replacement set make up the elements of the solution set of
the given equation. Only those values from the replacement set that make the open sentence true will
become elements of the solution set.
Even though you have an equation that has a solution set, if the value that would make the equation
true is not in your give domain you cannot say the equation is true.
Example: 2x + 2 = 4
Solution set would be {1}, If your replacement set was {3, 4, 5} then we would say that the equation is
false. In this example your solutions set would be {} (empty set) because the actual solution for the
equation was not contained in your replacement set.
Chapter 2.2 – Addition and Subtraction Equations
An addition equation is an equation that contains only the operation of addition.
Example: d + 5 = 14.
To solve an addition equation we need to find all the values of d such that the open sentence d + 5 = 14
is true.
Addition and subtraction are inverse operations. To solve an addition equation we will use the
Subtraction property of equality which says:
For real numbers a, b, and c, if a = b then a - c = b - c. Hence whatever you subtract from one side of
an equation you must subtract from the other side of the equation in order to keep both sides of the
equation equal.
When solving equations always check that your solution set makes your original equation true.
Substitute each value for the variable into the original equation to verify the solution makes your
equation true.
A subtraction equation is an equation that contains only the operation of subtraction. To solve a
subtraction equation we will used the Addition property of equality which states:
For real numbers a, b and c if a=b then a + c = b + c
Again you will need to substitute in the values you get for a solution set into the original equation to
determine if your answer is correct.
The = sign in an equation is the bridge. The unknown should stay on one side of the bridge.
Everything else must cross to the other side of the bridge. In order for a number to cross to the other
side of the bridge it must change signs (+) becomes (-) or (-) becomes (+).
Chapter 2.3 - Solve Multiplication and Division Equations
A multiplication equation is an equation that contains only one operation – multiplication.
Since our goal when solving equations is always to isolate the variable, in order to solve a
multiplication equation you have to use the Division Property of Equality which states:
You can divide one side on an equation by a real number as long as you divide the other side of the
equation by the same real number.
For real numbers a, b, and c if a = b then a/c = b/c as long as c does not equal 0.
We use division to solve a multiplication equation because division is the inverse operation to
multiplication.
Make sure when solving equations that you check your solution by plugging your solution back into
your original equation.
An equation with only one operation, division, is called a division equation. To solve a division
equation you have to use the inverse operation to division which is multiplication. You use the
Multiplication Property of Equality which states:
If you multiply one side of an equation by a real number you must also multiply the other side of the
equation by the same real number. If a, b and c are real numbers and a = b then a·c = b·c.
When you divide fractions you multiply by the reciprocal, so when the coefficient in front of a variable
is a fraction you just have to multiply both sides of the equation by the reciprocal of the fraction.
You can use the multiplication property of equality to solve multiplication equations so long as
you multiply by the reciprocal of the coefficient.
Chapter 2.4 – Solve Equations with Two Operations
We will isolate our variables by using inverse operations. We will sort of work out order of operations
backwards. First we will use our addition or subtraction properties of equality. When we have just a
variable and a coefficient on one side of the equation and constants on the other we will then use our
multiplication and division properties of equality.
Once we have our solution once again we need to check our answer.
Example: 11 – b = 32.
You can solve this equation in more than one way. You can subtract 11 from both sides and then divide
by -1, or you can add b to both sides and subtract 32 from both sides. Adding b to both sides will
make b positive.
When solving an equation in two operations if there happens to be several fractions you can compute
the solution to the equation by leaving the fractions and following your rules for operations of
fractions.
Or
You can multiply both sides of the equation by the LCD and work with whole numbers. You can do
this because we can take advantage of the multiplication property of equality.
If the unknown appears in more than one term, you need to combine like terms before you begin to
solve the equation.
Chapter 2.5- 2.5A – Solve Multi-Step Equations and Equations with letter coefficients
Nick, Joan and Maya selling pottery pieces.
n = number of pieces Nick has sold
n + 10 = number of pieces Joan has sold
2(n + 10) = pieces Maya has sold
2(n+10) = 40 number of pieces Maya has sold.
To isolate the variable and solve for n we first need to use the distributive property to get rid of the
grouping symbols in our equation.
2(n) + 2(10) = 40 distributive property
2n + 20 = 40 simplify with multiplication
Next we need to use the subtraction property of equality to get the term with the variable alone on
one side of the equation.
2n + 20 – 20 = 40 – 20
2n = 20 Simplify
Lastly we need to use the division property of equality to isolate the variable, n
2n/2 = 20/2
n = 10 simplify
Nick sold 10 pieces of pottery
Joan sold 20 pieces of pottery
Maya sold 40 pieces of pottery
Sometimes we will have an equation with a variable on both sides of the equation. We will then have
to use our subtraction and addition properties of equality to get the terms with the variables on one side
of the equation and the terms that are just constants on the other side of the equation.
A helpful hint is to collect all the terms with the variable on the side of the equation where the resulting
coefficient will be positive. You do not have to do this, however, it will prevent having to divide by a
negative number and the possibility of forgetting your sign.
Sometimes when we solve equations there may be one solution, infinitely many solutions, or no
solutions.
When a solution ends with a true numerical statement Example: 3 = 3, the original equation is called an
identity, and its solution set is the set of real numbers. {x| x is any real number}
When a solution process ends with a false numerical statement Example 7 = -11, the original equation
is called a contradiction, and its solution set is the null set. {}
When you are solving an equation for a variable, the solution can be written in terms of another
variable. If I want to solve the equation: ax + b = cx + d for the variable x, you follow the properties
of real numbers and treat the variables b, c and d as if they were numeric.
ax + b = cx + d
ax + b – b = cx + d – b
Subtraction property of equality
ax + cx = cx + d – b – cx
Subtraction property of equality
ax + cx = d – b
x(a + c) = d – b
Distributive property
x(a + c) = d – b
(a + c) (a + c)
Division property of equality
x=d–b
a+c
Chapter 2.6 – Solve Absolute Value Equations
The late night news reported that the day’s temperature varied 18 degrees from the midmorning reading
of 42 degrees. What were the day’s possible maximum and minimum temperature.
We do not know from this problem if the day’s temperature went up from the midmorning reading, or if
the temperature went down from the midmorning reading that is why we can have a possible maximum
or minimum temperature.
If the temperature went up so that 42 degrees was the minimum temperature, then the maximum
temperature was 18 degrees greater than 42.
If the temperature went down so that 42 was the maximum temperature, then the minimum temperature
was 18 degrees less than 42 degrees.
We can represent these two scenarios with just one absolute value equation.
| t – 42 | = 18
We have two points on the number line that can show a distance of 18 units away from 0 on the number
line, they are 18 and -18.
When solving absolute value equations we always have two cases for a solution:
Case 1: |x| = a
x = a when x is non-negative
Case 2: |x| = a
-x = a when x is negative
To solve an absolute value equation you want to isolate the absolute value expression on one side of the
equation.
In our example: Case 1: t – 42 = 18
+ 42 +42
---------------------t = 60
Case 2: -(t – 42) = 18
-t + 42 = 18
- 42 -42
-------------------t = -24
t = 24
The possible maximum temperature was 60 degrees and the possible minimum temperature was 24
degrees. You can check your solution in the original absolute value equation. | 60 – 42 | = 18 and
|24 – 42 | = 18.
In absolute value equations the variable has two possible values. It cannot be both at the same time – it
is one value or the other value.
Example 1 : 3 |y – 2 | - 4 = 11 The trick is to isolate the absolute value expression first.
Not all absolute value equations have two solutions. If an absolute value expression = 0 then there is
only one solution. Example: | x – 8 | + 20 = 20
If an absolute value expression corresponds to a negative value then there are no solutions.
Example: | y + 2 | + 7 = 1
Chapter 2.7 - Formulas and Literal Equations
A formula is an equation that states a rule for relationship among particular quantities.
Examples: Distance = rate * time (d = r•t), Fahrenheit to Celcius ( F= 9/5C + 32)
To rewrite the distance equation to develop a formula for the rate we have to isolate the variable we
want to solve for. So if d = rt then we want to use the division property of equality and divide both
sides by the variable t. If we do this then r = d/t.
This is called solving for a variable. You handle it the same way you handle solving equations where
the expectation is to isolate the variable to get a solution set. When working with formulas you are not
looking for a numerical answer what you are looking for is a rewritten equation. The trick is to always
remember what variable you want isolated. We will use our same properties to rearrange formulas that
we used to solve equations.
A formula is a type of literal equation which is an equation with two or more variables. In a given
literal equation you can solve for any one of the variables in terms of the others.
Let's try some:
Solve for d: 3(d+m) = 8
distribute first : 3d + 3m = 8
Subtraction property of equality: 3d + 3m – 3m = 8 – 3m
3d = 8 – 3m
Division property of equality:
3d/3 = (8 – 3m)/3
d = (8-3m)/3
d = 8/3 – m
Solve for z: 12z – a = 3z + 7a
addition property of equality:
Subtraction Property of equality
Division Property of equality
12z – a + a = 3z + 7a + a
12z = 3z + 8a
12z – 3z = 3z + 8a – 3z
9z = 8a
9z/9 = 8a/9
z = 8a/9
We have to remember what variables represent; they represent real numbers, so we can use the
properties of real numbers in order to evaluate literal equations for a particular variable. Realize in the
end that the solution you get will not be a set of real numbers, but a newly formulated equation.
Chapter 2.7 A – Dimensional Analysis
Dimensional analysis is the use of units of measurement to guide the solving of problems.
Dimensional analysis is a necessity for high school and college science courses.
An example of a problem requiring the use of dimensional analysis is as follows:
Quentin Quickness ran run 12 miles per hour. How many feet can he run per second?
Some useful conversion ratios that may help you solve the above problem are:
1 mile
5280 ft
1 minute
60 seconds
1 hour
60 minutes
Before we can solve the above problem we will work with an easier problem. Let’s begin with a
review.
A ratio is the amount of one quantity compared to the amount of another quantity. Written 3 ways; with
a colon (4:3), in fraction form (4/3), using word form (4 to 3).
Rate is a ratio that compares different units. Examples:
4 dogs
/3 cats or 40 miles/2 gallons.
Unit Rate is a rate which has been simplified to obtain one in the denominator. 40 miles/2 gallons is not a unit
rate because it has a 2 in the denominator. 20 mile/1 gallon is a unit rate.
Using dimensional analysis we will convert one measure or rate to an equivalent measure or rate (with
a different set of dimensions or units) by multiplying by one or more conversion ratios.
Let’s begin with an example. How many inches are in 7 feet?
We will begin with
What we are given
Use Ratios to get you from
given to where you are going
GIVEN: 7 feet
RATIO: 12 inches/1 foot
Solve, making sure the units cancel correctly:
7 feet x 12 inches = 84 = 84 inches
1 foot
1
There are 84 inches in 7 feet.
What you want to get
_______ number of inches
Now let’s try the above example about Quentin Quickness who runs 12 miles per hour. We want to
know how many feet he runs in a second. There are two conversions that have to take place. One
being the distance traveled from miles to feet and one being time spent running from hours to seconds.
GIVEN: 12 miles/1 hour
12 miles • 1 hour •
1 hour
60 minutes
RATIOS: 1 hr/60 min, 1 min/60 sec, 5280 ft/1 mile WANT: ____ ft/sec
1 minute •
60 seconds
5280 feet = 12 • 1 • 1 • 5280 feet = 17.6 feet
1 mile
1 • 60 • 60 • 1 second
1 second