1 2.1 Combining Like Terms Let’s review some terminology: Variables-Letters that represent numbers Expressions-collection of numbers, variables, grouping.

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2.1 Combining Like TermsLet’s review some terminology:

Variables-Letters that represent numbers

Expressions-collection of numbers, variables, grouping symbols, and operations symbols

Examples of Expressions:

5 4x-3 2(x+5)

Terms-are separated by addition or subtraction signs

3x – 6 terms are 3x and -6

Numerical Coefficient-is the number in front of the variable

in 4x the coefficient is 4

in x the coefficient is 1

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3x62 x

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2.1 Combining Like TermsConstant Term -a number term without a variable

in 4x + 5 the constant is 5

Like Terms – have same variables and the same exponents on those variables

3x and x

5xy and 2xy

5 and -3

223 yy

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Remember the distributive property? a(b+c) = ab + ac

We will use this to eliminate parenthesis by distributing and combining like terms5(x + 4) -2(3 - x) + 7 5 – (3x – 4)5x + 20 -6 + 2x + 7 5 – 1(3x - 4)

2x + 1 5 – 3x + 49 – 3x

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2.1 Combining Like TermsTo simplify an expression, use the distributive property to

eliminate parenthesis and then combine like terms.

Remember Factors are multiplied; Terms are added

You cannot solve expressions. You can only simplify them.

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2.2 The Addition Property of EqualityExpressions set equal to each other are called equations.

Solution-when a solution is substituted for the variable in an equation it makes a true statement. To check a possible solution, substitute it into the equation and see if it is true or not.

Addition Property of Equality

If a = b then a + c = b + c

If you have two things that are equal, you can add the same thing to both sides and the two things will remain equal.

We use this to solve equations-to isolate the variable:

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2.2 The Addition Property of Equality

• X + 8 = 15

+ -8 + -8

x = 7

• what could you add to both sides of the equation to isolate the variable?

• In other words, what would create a zero where the 8 is?• Add a -8 to both sides• This eliminates the constant (8) and isolates the variable• Your solution is 7

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2.3 The Multiplication Property of Equality

If a = b then ac = bc

If you have two things that are equal, you can multiply both sides by the same thing and the two things will remain equal.

We use this to solve equations also:

Remember that a reciprocal is the same as the multiplicative inverse. 3 1/3 ½ 2

You can isolate the variable by multiplying by the reciprocal of the coefficient OR by dividing by the coefficient itself.

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2.3 The Multiplication Property of Equality

dividing by the coefficient itself

3x = 24

3 3

X = 8

multiplying by the reciprocal of the coefficient

3x = 24

(3x) = (24)

x = 8

3

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1

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2.4 Solve Equations - variable on one sideThe following steps will work for all equations.Isolate the variable using the following steps:1. Use distributive property to remove parenthesis

2(x + 2) = 2x + 42. Clear fractions by multiplying both sides by the LCD

(optional)3. Combine Like Terms on each side of equation4. Use addition property of equality to get all the variable

terms on one side of the equation and all the constant terms on the other side

5. Use multiplication property of equality to get rid of the coefficient on the variable

6. Check your solution (optional)

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2.4 Solve Equations - variable on one sideRemember these terms:Evaluate-find a numerical value

3(4) – 9 (1-3) 5Simplify-perform the operations and combine like terms

2(x – 5) + 7 (x+ 2)Solve-find the value that makes the equation true

2x + 7 = 15-7 -72x = 8 2 2x = 4

Check-substitute the value into the original equation to see if it is true: 2x + 7 = 152(4) + 7 = 158 + 7 = 15 15 = 15

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2.5 Identities and Contradictions

2(x + 3) = 2x + 6

2x + 6 = 2x + 6

-2x -2x

6 = 6

-6 -6

0=0

Distribute

Both sides match

Everything drops out

Solution:

Any real number

Identities-true for an infinite number of solutions

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2.5 Identities and Contradictions

2(x + 5) = 2x + 6

2x + 10 = 2x + 6

-2x -2x

10 = 6

-6 -6

4=0

Distribute

Variable terms match

Variable drops out

What you are left with is NOT true

There is NO solution

Contradictions- have no solution

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2.6 FormulasFormulas are commonly used equations that

express a specific mathematic relationship

Examples: A = l x w or i = prt

Evaluate-means to substitute in the given numeric values for the appropriate variables; perform the indicated operations. Find a “value.”

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2.6 Formulas-simple interesti = prt (interest = principal x rate x time)

Interest=money earned by an investment OR money charged for borrowing money

Principal=the amount of money invested or borrowed

Rate=the percentage rate expressed as a decimal ( 5% = 0.05 or 7.5% = 0.075 )

Time=the time the money was invested or borrowed IN YEARS (3 months = ¼ year)

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2.6 Formulas-simple interestIf I borrow $10,000 for 3 years at a 5% APR,

how much will I pay in interest?

i = p r t

i = 10,000 x 0.05 x 3

i = 1500

$1500 will be paid in simple interest

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2.6 Formulas-simple interestIf I invest $7,000 for 10 years at an unknown

interest rate, and earn $4200 in interest, what was the interest rate that I earned?

i = p r t

4200 = 7,000 x r x 10

4200 = 70,000 x r

70,00070,000

0.06 = r

The rate of return was 6 %.

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2.6 Formulas-geometric applicationsYou will find a variety of geometric formulas in your

book. One chart gives the 2-d formulas for perimeter and area. Another gives the formulas for a circle (circumference and area), and a third gives the 3-d formulas. You do not need to memorize these formulas, but you do need to know how to use them.

Remember:Perimeter is the distance around a 2-d object (like

fencing a yard);measured in regular unitsArea is the amount of space inside a 2-d object

(like painting a wall); measured in square units

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2.6 Formulas-geometric applicationsThe Monesteros have an in-ground pool that they

would like to fence in. The area needing fencing is 30 feet by 50 feet – rectangular. Find the amount of fence required to enclose all four sides of the pool area.

P = 2 L + 2 WP = 2 (50) + 2 (30)P = 100 + 60P = 160The perimeter or amount of fence needed is 160

feet.

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2.6 Formulas-geometric applicationsThe Monesteros have an in-ground pool that they

would like to fence in. The area needing fencing is 30 feet by 50 feet – rectangular. Find the area that is now enclosed by the new fence.

A = L x W

A = 50 x 30

A = 1500

The area inside the new fence is 1500 sq feet.

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2.6 Formulas-geometric applications

The Monesteros recently experienced an alien space craft landing in their yard. The diameter of the ring left in the grass is six meters. Find the circumference and area of the ring.

C = 2 π r A = π

C = 2 x 3.14 x 3 A = 3.14 x 3 x 3

C = 18.85 meters A = 28.3 sq m

2r

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2.6 Formulas-solve for a specified variableSolve for W Solve for tP = 2L + 2W i = p r t-2L -2L i = p r tP-2L = 2W pr pr 2 2P-2L = W i = t 2 prP – L = W2

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2.6 Formulas-solve for a specified variable

Solve for y

2x + 3y = 12

-2x -2x

3y = 12 – 2x

3 3

y = 4 – 2x

3

y = 12 – 2x

3

Given a value for x, find y in the previous example.

If x = 0, then

y = 12 – 2(0) = 12 =4

3 3

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2.8 Inequalities in One Variable

To solve inequalities, we use the same properties that we used to solve equations EXCEPT when you are eliminating the coefficient, if you multiply or divide by a negative number, you must FLIP the inequality sign.

- 3x – 5 10

+5 +5

-3x 15

-3 -3

x -5

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2.8 Inequalities in One VariableTo graph the solution of an inequality, we use a number line

and we shade according to the inequality sign. For example: strictly less than and strictly greater than

signs will use an open circle to indicate that the point itself is not included in the graph.

Less than or equal to and greater than or equal to signs will use a closed circle to indicate that the point itself is included in the graph.

In general, if the inequality is written with the variable on the left, the arrow of the inequality sign will indicate the direction you need to shade. (less than to the left; greater than to the right)

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2.8 Inequalities in One Variable

closed circle < > open circle

Remember our solution from the example we did?

X ≥ -5

Let’s graph that on a number line: