Download Tool Box - Unit 1 Relationships between Quantities Reasoning with

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Algebra I Tool Box: Unit 1- Relationships between
Quantities & Reasoning with Equations
a
Three Star Questions (
):

What is the inverse operation of addition?

What is the inverse operation of division?



In order to solve a proportion you need to…
What happens to the inequality sign if you multiply or divide by a negative number?
What is the solution when the variables are eliminated (cancelled) and the inequality is
not true?
What is the solution when the variables are eliminated (cancelled) and the inequality is
true?

Definitions:
Vocab
Exponent
Simplify
Evaluate
Expression
Equation
Equivalent
Equations
Isolate
Inverse
Operations
Define it
Example
Your Reminder
Identify
Ratio
Literal Equation
Solution of an
Inequality
Equivalent
Inequalities
Concepts/Properties:
Order of Operations:
Re-write in your own words:
1. Perform any operation(s) inside grouping symbols,
such as parentheses ( ) and brackets [ ]. A fraction bar
also acts as a grouping symbol.
2. Simplify powers.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.
Ex 1: What is the simplified form of the expression?
6  23  2
6  23  2
3
 4   2
 4  4  4  2
 64  2
 32

Simplify in grouping symbols
1st.

Simplify the power of 3.

Perform multiplication inside
grouping symbol.

Perform multiplication to get
64

Perform division to simplify to
answer.
Addition Property of Equality: Adding the same
Re-write in your own words:
number to each side of an equation produces an
equivalent equation.
Algebraic
For any real numbers a, b, and c, if a = b,
then a + c = b + c.
Ex 2:
x 3  2
x 33  2 3
x 3  2
x 33  2 3
x05
x 5
In order to isolate the
variable, we need to use the
addition property of
equality. By adding 3 to both
sides of the equation, we
accomplish our goal of
getting the variable by
itself. In this case, x is equal
to 5.
Ex 3: Create your own example that’s different from
the one above but still uses the Addition Property of
Equality.
Subtraction Property of Equality: Subtracting
Re-write in your own words:
the same number to each side of an equation produces
an equivalent equation.
Algebraic
For any real numbers a, b, and c, if a = b,
then a - c = b - c.
Ex 4:
x47
x44 74
x47
x44 74
x03
x 3
In order to isolate the
variable, we need to use the
subtraction property of
equality. By subtracting 4 to
both sides of the equation,
we accomplish our goal of
getting the variable by
itself. In this case, x is
equal to 3.
Ex 5: Create your own example that’s different from
the one above but still uses the Subtraction Property
of Equality.
Multiplication Property of Equality: Multiplying
Re-write in your own words:
each side of an equation by the same nonzero number
produces an equivalent equation.
Algebraic
For any real numbers a, b, and c, if a = b,
then a * c = b * c.
Ex 6:
x
2
3
x
3  23
3
x
2
3
x
3  23
3
x6
In order to isolate the
variable, we need to use the
multiplication property of
equality. By multiplying 3 to
both sides of the equation
we accomplish our goal of
getting the variable by
itself. In this case, x is equal
to 6.
Ex 7: Create your own example that’s different from
the one above but still uses the Multiplication Property
of Equality.
Division Property of Equality: Dividing each side
Re-write in your own words:
of an equation by the same nonzero number produces
an equivalent equation.
Algebraic
For any real numbers a, b, and c, such that c ≠ 0,
if a = b, then
Ex 8:
a b
 .
c c
5x  20
5 x 20

5
5
5x  20
5 x 20

5
5
x4
In order to isolate the
variable, we need to use the
division property of equality.
By dividing 5 to both sides
of the equation we
accomplish our goal of
getting the variable by
itself. In this case, x is equal
to 4.
Ex 9: Create your own example that’s different from
the one above but still uses the Division Property of
Equality.
Cross Product Property of a Proportion: The
Re-write in your own words:
cross products of a proportion are equivalent.
Algebraic
If
a c
 , where b ≠ 0 and d ≠ 0, then a * d = b * c.
b d
x 9

4 12
Ex 10:
x 9

4 12
12  x  4  9
In order to isolate the variable, we
need to use the cross product
property of a proportion. By taking
12 x  36
the cross product, we get an
12 x 36

12 12
the division property of equality,
x 3
equation with one step. By using
we accomplish our goal of isolating
the variable. In this case, x is
equal to 3.
Ex 11: Create your own example that’s different from
the one above but still uses the Cross Product Property
of a Proportion.
Addition Property of Inequality: Adding the
Re-write in your own words:
same number to each side of an inequality produces an
equivalent inequality.
Algebraic
Let a, b, and c be real numbers.
If a > b, then a + c > b + c.
If a < b, then a + c < b + c.
This property is also true for ≥ and ≤.
Ex 12:
x 3 2
x 33  2 3
x 3 2
x 33  2 3
x05
x 5
In order to isolate the
variable, we need to use the
addition property of
inequality. By adding 3 to
both sides of the inequality,
we accomplish our goal of
getting the variable by
itself. In this case, x is
great than 5.
Ex 13: Create your own example that’s different
from the one above but still uses the Addition Property
of Inequality.
Subtraction Property of Inequality:
Re-write in your own words:
Subtracting the same number to each side of an
inequality produces an equivalent inequality.
Algebraic
Let a, b, and c be real numbers.
If a > b, then a - c > b - c.
If a < b, then a - c < b - c.
This property is also true for ≥ and ≤.
Ex 14:
x47
x4474
x47
x4474
x03
x3
In order to isolate the
variable, we need to use the
subtraction property of
inequality. By subtracting 4
to both sides of the
inequality, we accomplish
our goal of getting the
variable by itself. In this
case, x is less than 3.
Ex 15: Create your own example that’s different
from the one above but still uses the Subtraction
Property of Inequality.
Multiplication Property of Inequality:
Re-write in your own words:
Multiplying each side of an inequality by the same
nonzero number produces an equivalent inequality.
Algebraic
Let a, b, and c be real numbers with c > 0.
If a > b, then a * c > b * c.
If a < b, then a * c < b * c.
Let a, b, and c be real numbers with c < 0.
If a > b, then a * c < b * c.
If a < b, then a * c > b * c.
Ex 16a:
x
2
3
x
3  23
3
 Isolate the variable by using the
multiplication property of inequality.
 Perform multiplication to simplify.
 x is less than or equal to 6.
x6
Ex 17: Create your own example that’s different
from the one above but still uses the Multiplication
Property of Inequality.
Ex 16b:
x
5
4
x
 4  5  4
4
x  20
 Isolate the variable by using the
multiplication property of
inequality.
 Perform multiplication to simplify.
 Switch the inequality symbol
 x is greater than or equal to -20.
Division Property of Inequality: Dividing each
Re-write in your own words:
side of an inequality by the same nonzero number
produces an equivalent inequality.
Algebraic
Let a, b, and c be real numbers with c > 0.
If a > b, then
If a < b, then
a b
 .
c c
a b
 .
c c
Let a, b, and c be real numbers with c < 0.
If a > b, then
If a < b, then
a b
 .
c c
a b
 .
c c
Ex 18a:
5x  20
5 x 20

5
5
 Isolate the variable by using the
division property of inequality.
 Perform division to simplify.
 x is less than 4.
x4
Ex 18b:
 2x  16
 2 x 16

2 2
x  8
 Isolate the variable by using the
division property of inequality.
 Perform division to simplify.
 Switch the inequality symbol
 x is greater than -8.
Ex 19: Create your own example that’s different
from the one above but still uses the Division Property
of Inequality.
Essential Questions:
How can we utilize equation to solve problems?
How do equations factor into the real world?
What types of relationships can be modeled by linear
How can we determine the solutions to an inequality?
equations?
In what ways can we use inequalities to write, solve, and
What is the difference between a compound inequality
model situations?
involving AND & OR statements?