Solving Linear Equations and Inequalities

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Solving Linear Equations and Inequalities

• Chapter 2

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Chapter Sections
2.1 Combining Like Terms 2.2 The Addition
Property of Equality 2.3 The Multiplication
Property of Equality 2.4 Solving Linear
Equations with a Variable on One Side of the
Equation 2.5 Solving Linear Equations with a
Variable on Both Sides of the Equation 2.6
Formulas 2.7 Ratios and Proportions
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2.1

• Combining Like Terms

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The parts in an algebraic expression that are
added are called the terms of the expression.
Terms
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The numerical part of a term is the numerical
coefficient or coefficient.
Terms
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Like terms are terms that have the same variables
with the same exponents.
Like Terms
Unlike Terms 20x, x2, x3 6xy, 2xyz, w2
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Combining Like Terms

• Determine which terms are like terms.
• Add or subtract the coefficients of the like
  terms.
• Multiply the number found in step 2 by the common
  variable(s).

Example 5a 7a 12a
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Distributive Property

• For any real numbers a, b, and c,
• a(b c) ab bc

Example 3(x 5) 3x 15 (This is not equal
to 18x! These are not like terms.)
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Simplifying an Expression

• Use the distributive property to remove any
  parentheses.
• Combine like terms.

Example Simplify 3(x y) 2y 3x 3y
2y (Distributive Property) 3x 5y (Combine
Like Terms) (Remember that 3x 5y cannot be
combined because they are not like terms.)
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2.2

• The Addition Property of Equality

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Linear Equations
A linear equation in one variable is an equation
that can be written in the form ax b c where
a, b, and c are real numbers and a ? 0.
The solution to an equation is the number that
when substituted for the variable makes the
equation a true statement.
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Solutions to Equations
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Equivalent Equations
Two or more equations with the same solution are
called equivalent equations.
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Addition Property of Equality
If a b, then a c b c for any real numbers
a, b, and c.
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2.3

• The Multiplication Property of Equality

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Multiplication Property of Equality

• If a b, then a c b c for any real numbers
  a, b, and c.

Example Solve the equation 12y 15.
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Multiplication Property of Equality
x -3
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2.4

• Solving Linear Equations with a Variable on One
  Side of the Equation

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Equations with a Variable on One Side

• If the equation contains fractions, multiply both
  sides by the LCD to eliminate fractions.
• Use the distributive property to remove
  parentheses.
• Combine like terms on the same side of the equal
  sign.
• Use the addition property to obtain an equation
  in the form ax b.
• Use the multiplication property to isolate the
  variable.
• Check the solution in the original equation.

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Solving Equations
Example Solve the equation 3(4 x) 5x 9.
12 3x 5x 9 (Distributive property)
12 2x 9 (Combine like terms.)
2x -3
Dont forget to check!
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Solving Equations with Fractions
12 15n 5 (Simplify.)
15n -7 (Add 12 to both sides.)
n -7/15 (Divide both sides by
15.)
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Helpful Hints and Definitions

• To evaluate an expression means to find its
  numerical value.
• To simplify an expression means to perform
  operations and combine like terms.
• To solve an equation means to find the value of
  the variable that makes the equation a true
  statement.

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2.5

• Solving Linear Equations with a Variable on Both
  Sides of the Equation

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Equations with a Variable on Both Sides

• If the equation contains fractions, multiply both
  sides by the LCD to eliminate fractions.
• Use the distributive property to remove
  parentheses.
• Combine like terms on the same side of the equal
  sign.
• Use the addition property to obtain an equation
  in the form ax b. (This may need to be done
  twice.)
• Use the multiplication property to isolate the
  variable.
• Check the solution in the original equation.

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Solving Equations
Example Solve the equation 12 2x 3(x 2)
4x 12 x.
12 2x 3x 6 4x 12 x (Distributive
Property)
6 5x 3x 12 (Combine like
terms)
6 8x 12 (Add 5x to
both sides)
6 8x (Subtract 12 from both
sides.)
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Solving Equations with Decimals
Example Solve the equation. 5(3.2x 3)
2(x 4)
16x 15 2x 8 (Distribute.)
14x 15 8 (Subtract 2x from both sides.)
14x 7 (Add 15 to both sides.)
x ½ (Divide both sides by 14.)
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Identities and Contradictions
Equations that have a single value for a solution
(like the kind solved so far) are called
conditional equations. Equations that are true
for infinitely many solutions are called
identities. Equations that have no solution are
called contradictions.
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2.6

• Formulas

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Simple Interest

• Interest principal rate time

Example Sally Simon decided to borrow 6000 from
her bank to help pay for a car. Her loan was for
3 years at a simple interest rate of 8. How
much interest will Sally pay?
Interest principal rate time
(6000)(0.08)(3)
1440
Sally will pay 1440 interest.
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Geometric Formulas

• The perimeter, (P) is the sum of the lengths of
  the side of a figure.

The area, (A) is the total surface within the
figures boundaries.
A quadrilateral is the general name for a
four-sided figure.
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s
w
l
h
w
l
b
a
h
c
d
c
h
a
b
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Geometric Formulas

• Example
• Mike Morgan has a rectangular lot that measures
  100 feet by 60 feet. If Mike wants to fence in
  his lot, how much fencing will he need? What is
  the area of the lot?

P 2(100) 2(60) 200 120 320 ft A lw
100(60) 6000 feet2
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Geometric Formulas

• The circumference, (C) is the distance around a
  circle.

The radius, (r) is the line segment from the
center of the circle to any point on the circle.
The diameter of a circle is a line segment
through the center whose endpoints both lie on
the circle.
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Geometric Formulas

• The value of pi, (?) is an irrational number.
  Pi is approximately equal to 3.14.

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Geometric Formulas

• Example
• A circular swimming pool has a diameter of 24
  feet. What is the circumference of the pool?

Remember that the radius is half of the diameter.
C 2? r 2 (3.14)(12) ? 75.4 The circumference
is approximately 75.4 feet.
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h
w
l
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Geometric Formulas

• Volume is measured in cubic units, such as cubic
  inches or cubic meters.

Example Bob has an empty oil drum that he uses
for storage. The drum is 4 feet high and has a
diameter of 24 inches. Find the volume of the
drum. (Hint 24 inches ? feet)
V ? r2h 3.14(1)2(4) ? 12.56 ft3
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2.7

• Ratios and Proportions

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Ratios
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Proportions
A proportion is a statement of equality between
two ratios.
a and d are the extremes b and c are the means
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Cross-Multiplication
(Note that the product of the means equals the
product of the extremes.)
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Solving with Proportions

• Understand the problem.
• Translate the problem into mathematical language
  by representing the unknown by a variable and
  setting up the proportion.
• Solve the proportion.
• Check your answer.
• Make sure you have answered the question.

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Solving with Proportions

• Example
• A gallon of paint covers 825 square feet. How
  much paint is needed to cover a house with a
  surface area of 5775 square feet?

Answer Let x number of gallons
1(5775) 825x (Cross multiply.)
7 x (Divide by 825.)
7 gallons are needed.