Math 131 - Chapter 2 Linear Equations, Inequalities, and Applications

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Math 131 - Chapter 2 Linear Equations,
Inequalities, and Applications

• 2-1 Linear Equations in One Variable
• 2-2 Formulae
• 2-3 Applications
• 2-5 Linear Inequalities in One Variable

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2-1 Linear Equations in One Variable

• A linear equation in one variable can be written
  in the form Ax B C, where A,B, and C are real
  numbers with A ? 0.
• Example 1 x 4 -2
• Note x 4 by itself is called an algebraic
  expression.
• Example 2 2k 5 10
• Note A linear equation is also called a first
  degree equation since the highest power of x is
  1.
• The following are called non-linear equations
• y 3x2 10

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2-1 Linear Equations in One Variable

• Decide whether a number is a solution of a
  linear equation.
• Example 3 8 is a solution of the equation x 3
  5
• An equation is solved by finding its solution
  set.
• The solution set of x 3 5 is 8.
• Note Equivalent equations are equations that
  have the same solution set
• Example 4 5x 2 17, 5x 15 and x 3 are
  equivalent.
• The solution set is 3.

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2-1 Linear Equations in One Variable

• Addition Properties of Equality
• For all real numbers A,B, and C, the equations A
  B and A C B C are equivalent.
• Note The same number can be added to each side
  of an expression without changing its solution
  set.
• Example 5 Solution of the equation x 3 5
• x 3 3 5 3
• x 8
• The solution set of x 3 5 is 8.
• Example 6 5x 2 17, 5x 2 (-2) 17
  (-2) and 5x 15 are equivalent.
• The solution set is 3.

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2-1 Linear Equations in One Variable

• Multiplication Properties of Equality
• For all real numbers A,B, and for
• C ? 0, the equations A B and AC BC are
  equivalent.
• Note Each side of an equation may be multiplied
  by the same non-zero number without changing its
  solution set.
• Example 7 Solution of the equation
• The solution set of 5x 15 is 3.

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2-1 Linear Equations in One Variable

• Example 8 Solve -7 3x 9x 12x -5
• -7 7 6x 12x 5 7
• -6x 12x 12x 12x 2
• -18x 2
• The solution set is .

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2-1 Linear Equations in One Variable

• Solving a Linear Equation in One Variable
• Step 1 Clear fractions. (Eliminate any fractions
  by multiplying each side by the least common
  denominator.)
• Step 2 Simplify each side separately. (Use the
  distributive property to clear parenthesis and
  combine like terms.)
• Step 3 Isolate the variable terms on the left
  side of the equal sign. (Use the addition
  property to get all the variable terms on the
  left side and all the numbers on the right side.)
  
• Step 4 Isolate the variable. (Use the
  multiplication property to get an equation where
  the coefficient of the variable is 1.)
• Step 5 Check. (Substitute the proposed solution
  into the original equation and verify both sides
  of the equal sign are equivalent.)

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2-1 Linear Equations in One Variable

• Example 8 Solve 6 (4 x) 8x 2(3x 5)
• Step 1 Does not apply
• Step 2 6 4 x 8x 6x 10
• -x 2 2x 10
• Step 3 -x (-2x) 2 (-2) 2x (-2x) 10
  (-2)
• -3x -12
• Step 4
• Step 5 6 (4 4) 8(4) 2(3(4) 5)
• 6 8 32 - 34
• -2 -2
• The solution set is 4.

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2-1 Linear Equations in One Variable

• Example 9 Solve
• Step 1
• Step 2
• Step 3
• Step 4
• Step 5
• The solution set is -1.

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2-1 Linear Equations in One Variable

• Identification of Conditional Equations,
  Contradictions, and Identities
• Some equations that appear to be linear have no
  solutions, while others have an infinite number
  of solutions.
• Note Identification requires knowledge of the
  information in the table.

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2-1 Linear Equations in One Variable

• Example 9 Solve each equation, Decide whether
  it is conditional, identity, or contradiction.
• A) 5x 9 4(x-3)
• 5x 9 4x 12
• x -3 ? conditional
• The solution set is -3
• B) 5x 15 5(x-3)
• 5x 15 5x 15
• 0 0 ? identity
• The solution set is all real numbers
• C) 5x 15 5(x-4)
• 5x 15 5x 20
• 0 -5 ? contradiction
• The solution set is or 0 (null set)

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2-2 Formulae

• A mathematical model is an equation or
  inequality that describes a real situation.
  Models for many applied problems already exist
  and are called formulas.
• A formula is a mathematical equation in which
  variables are used to describe a relationship.
• Common formulae that will be used are
• d rt distance (rate)(time)
• I prt Interest (principal)(rate)(time)
• P 2L 2 W Perimeter 2(Length) 2(Width)
• Additional formulae are included on the inside
  covers of your text.

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2-2 Formulae

• Solving for a specified variable
• Step 1 Get all terms containing the specified
  variable on the left side of the equal sign and
  all other terms on the right side.
• Step 2 If necessary, use the distributive
  property to combine the terms with the specified
  variable.
• Step 3 Isolate the variable. (Use the
  multiplication (division) property to get an
  equation where the coefficient of the variable is
  1.)
• Example 1 Solve m 2k 3b for k
• 2k m 3b
• k (m 3b)/2

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2-2 Formulae

• Example 2 Solve
• Example 3 Solve

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2-2 Formulae

• Example 4 Given a distance of 500 miles and a
  rate (speed) of 25 mph, find the time for the
  trip.
• Example 5 A 20 oz mixture of gasoline and oil
  contains 1 oz of oil. What percent of the mixture
  is oil?

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2-3 Applications of Linear Equations

• Translating words to mathematical expressions.
• Note There are usually key words or phrases
  that translate into mathematical expressions.

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2-3 Applications of Linear Equations

• Translating words to mathematical expressions.
• Note There are usually key words or phrases
  that translate into mathematical expressions.

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2-3 Applications of Linear Equations

• Translating words to mathematical expressions.
• Note There are usually key words or phrases
  that translate into mathematical expressions.

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2-3 Applications of Linear Equations

• Translating words in to equations.
• Note The symbol for equality, , is often
  indicated by the word is.

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2-3 Applications of Linear Equations

• Distinguishing between expressions and
  equations.
• Note An expression is a phrase. An equation
  includes the symbol and translates the
  sentence.
• Example 1 Decide whether each is an expression
  or an equation.
• A) 5x 3(x2) 7 Equation
• B) 5x 3(x2) Expression

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2-3 Applications of Linear Equations

• Solving Applied Problems.
• Note The following six steps are useful in
  developing a technique for solving applied
  problems.
• Step 1 Read the problem carefully until you
  understand what is given and what is to be found.
  
• Step 2 Assign a variable to represent the
  unknown value, using diagrams or tables as
  needed.
• Step 3 Write an equation using the variable
  expressions.
• Step 4 Solve the equation. (Isolate the variable
  on the left side of the equals sign)
• Step 5 State the answer with appropriate units.
  (Does it seem reasonable?)
• Step 6 Check the answer.

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2-3 Applications of Linear Equations

• Solving Applied Problems.
• At the end of the 1999 baseball season, Sammy
  Sosa and Mark McGwire had a lifetime total of 858
  homeruns. McGwire had 186 more than Sosa. How
  many runs did each player have?
• Step 1 Total runs 858 Sosa has 186 less
  than McGwire
• Step 2 Let x Sosas runs, then x 186
  McGwires runs
• Step 3 x (x 186) 858
• Step 4 2x 186 858
• 2x 672
• x 336 x 186 522
• Step 5 Sosa had 336 homeruns. McGwire had 522
  homeruns.
• Step 6 336 522 858
• 858 858
• Checked

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2-3 Applications of Linear Equations

• Solving Applied Problems.
• A woman has 34,000 to invest. She invests some
  at 17 and the balance at 20. Her total annual
  interest income is 6425. Find the amount
  invested at each rate.
• Step 1 Total investment 34,000 Total
  interest earned is 6245
• Step 2 Let x amount at 17, then 34,000 - x
  amount at 20
• Step 3 .17x .20(34,000 - x) 6245
• Step 4 .17x 6800 - .20x 6245
• -.03x -555
• x 18,500 34,000 18,500 15,500
• Step 5 18,500 invested at 17 15,500 invested
  at 20
• Step 6 .17(18,500) .20(15,5000 6245
• 3145 3100 6245
• 6245 6245
• Checked

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2-3 Applications of Linear Equations

• Solving Applied Problems.
• How many pounds of candy worth 8 per lb should
  be mixed with 100 lb of candy worth 4 per lb to
  get a mixture that can be sold for 7 per lb?
• Step 1 Total mixture should sell for 7/lb.
  Given 100 lbs at 4/lb and an unknown amount at
  8/lb.
• Step 2 Let x amount at 8 then x 100 should
  sell at 7/lb
• Step 3 8x 4(100) 7(x 100)
• Step 4 8x 400 7x 700
• x 300
• Step 5 300 lb at 8/lb
• Step 6 8(300) 4(100) 7(300 100)
• 2400 400 7(400)
• 2800 2800
• Checked

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2-3 Applications of Linear Equations

• Solving Applied Problems.
• How much water must be added to 20 L of 50
  antifreeze solution to reduce it to a 40
  antifreeze solution?
• Step 1 Total mixture should be 40
  antifreeze. Given 20 L at 50 produces 10 L of
  water and 10 L of antifreeze. Add an unknown
  amount of water.
• Step 2 Let x amount of water added, then x
  10 will be the water in the final solution and
  the amount of antifreeze is 10 L.
• Step 3
• Step 4
• Step 5 5 L of water
• Step 6 Check
• Checked

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2-5 Linear Inequalities in One Variable

• Interval Notation is used to write solution sets
  of inequalities.
• Note A parenthesis is used to indicate an
  endpoint in not included. A square bracket
  indicates the endpoint is included.

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2-5 Linear Inequalities in One Variable

• Interval Notation is used to write solution sets
  of inequalities.
• Note A parenthesis is used to indicate an
  endpoint in not included. A square bracket
  indicates the endpoint is included.

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2-5 Linear Inequalities in One Variable

• Interval Notation is used to write solution sets
  of inequalities.
• Note A parenthesis is used to indicate an
  endpoint in not included. A square bracket
  indicates the endpoint is included.

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2-5 Linear Inequalities in One Variable

• A linear inequality in on variable can be
  written in the form Ax B C, where A, B, and C
  are real numbers with A ? 0.
• Note The next examples include definitions and
  rules for lt , gt, ?, and ? .
• Examples of linear inequalities
• x 5 lt 2 x 3 gt 5 2k 4 ? 10

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2-5 Linear Inequalities in One Variable

• Solving linear inequalities using the Addition
  Property
• For all real numbers A, B, and C, the
  inequalities A lt B and A C lt B Ca are
  equivalent.
• Note As with equations, the addition property
  can be used to add negative values or to subtract
  the same number from each side .
• Example 1 Solve k 5 gt 1
• k 5 5 gt 1 5
• k gt 6
• Solution set (6,?)
• Example 2 Solve 5x 3 ? 4x 1 and graph the
  solution set.
• 5x 4x ? -1 3
• x ? -4
• Solution set -4,?)

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2-5 Linear Inequalities in One Variable

• Solving linear inequalities using the
  Multiplication Property
• For all real numbers A, B, and C, with C ? 0 ,
• 1) if C gt 0, then the inequalities A lt B and AC
  lt BC are equivalent.
• 2) if C lt 0, then the inequalities A lt B and AC
  gt BC are equivalent.
• Note Multiplying or Dividing by a negative
  number requires the inequality sign be reversed.
• Example 1 Solve -2x lt 10
• x gt -5
• Solution set (-5,?)
• Example 2 Solve 2x lt -10
• x lt -5
• Solution set (-?,-5)
• Note The first example requires a symbol change
  because both sides are multiplied by (-1/2). The
  second example does not because both sides are
  multiplied by (1/2)

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2-5 Linear Inequalities in One Variable

• Solving linear inequalities using the
  Multiplication Property
• Example 3 Solve 9m lt -81 and graph the solution
  set
• m gt 9
• Solution set (9, ?)

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2-5 Linear Inequalities in One Variable

• Solving a Linear Inequality
• Step 1 Simplify each side separately. If
  necessary, use the distributive property to clear
  parenthesis and combine like terms.
• Step 2 Use the addition property to get all
  terms containing the specified variable on the
  left side of the inequality sign and all other
  terms on the right side.
• Step 3 Isolate the variable. (Use the
  multiplication (division) property to get an
  inequality where the coefficient of the variable
  is 1.)
• Example 4 Solve 6(x-1) 3x ? -x 3(x 2) and
  graph the solution set
• Step 1 6x-6 3x ? -x 3x 6
• 9x - 6 ? 4x 6
• Step 2 13x ? 0
• Step 3 x ? 0
• Solution set 0, ?)

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2-5 Linear Inequalities in One Variable

• Solving a Linear Inequality
• Example 5 Solve
  and graph the solution set
• Step 1
• Step 2
• Step 3
• Solution set , ?)

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2-5 Linear Inequalities in One Variable

• Solving applied problems using linear
  inequalities
• Example 6 Solve 5 lt 3x 4 lt 9 and graph the
  solution set
• 5 4 lt 3x lt 9 4
• 9 lt 3x lt 13
• 3 lt x lt (13/3)
• Solution set ( 3, )

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2-5 Linear Inequalities in One Variable

• Solving applied problems using linear
  inequalities
• Note Expressions for inequalities sometimes
  appear as indicated in the table
• Example 7 Teresa has been saving dimes and
  nickels. She has three times as many nickels as
  dimes and she has at least 48 coins. What is the
  smallest number of nickels she might have?
• Let x equal the number of dimes
• Then 3x is the number of nickels
• and 3x x ? 48
• 4x ? 48
• x ? 12 ? 36 nickels

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2-5 Linear Inequalities in One Variable

• Solving applied problems using linear
  inequalities
• Example 8 Michael has scores of 92, 90, and 84
  on his first three tests. What score must he get
  on the 4th test in order to maintain an average
  of at least 90?
• Let x equal the 4th score