1.7 Solving Inequalities

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1.7 Solving Inequalities

• Algebra II
• Mrs. Spitz
• Fall 2006

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Lesson Objectives

• After studying this lesson, you should be able
  to
• Solve an inequality and graph the solution set,
  and
• Use inequalities to solve problems

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Assignment

• Pg. 39 11-32 and 38
• EC problems 39-45

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Application

• Jose and Kyle are soccer players on the Taos High
  School team. If you compare their scoring for
  the season, only one of the following statements
  will be true.
• Jose scored fewer goals than Kyle.
• Jose scored the same number of goals as Kyle.
• Jose scored more goals than Kyle

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Application

• Let j represent the number of goals Jose scored
  and k represent the number of goals Kyle scored.
  You can compare the scoring using an inequality
  or an equation.
• j lt k j k j gt k
• This is an illustration of the trichotomy
  property.

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Trichotomy Property

• For any two real numbers, a and b, exactly one of
  the following statements is true
• a lt b a b a gt b

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Addition and Subtraction properties for
Inequalities

• Adding the same number to each side on an
  inequality does not change the truth of the
  inequality.
• For any real numbers, a, b, and c
• 1. If agtb, then a c gt b c and a c gt b c
• 2. If altb, then a clt b c and a c lt b c
• These properties can be used to solve an
  inequality. The solution set of an equality can
  be graphed on a number line.

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Example 1 Solve 8x 5 lt 7x - 3

• 8x 5 lt 7x 3
• -7x 8x 5 lt -7x 7x 3 Add -7x to each side
• x 5 lt -3
• x 5 (-5) lt -3 (-5) Add -5 to each side
• x lt -8

A circle means this point is NOT included.
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Example 1 Solve 8x 5 lt 7x - 3

• Any real number less than -8 is a solution.
  To check, substitute -8 for x in the inequality.
  The two sides should be equal. Then substitute a
  number less than -8. The inequality should be
  true, or if it isnt you made a math error.

A circle means this point is NOT included.
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NoteYou know that 15 gt -6 is a true inequality.
What happens if you multiply the numbers on each
side by a positive number or a negative number?
Is it still true?

• 15 gt -6
• 7(15) gt 7(-6)
• 105 gt -42 true
• Multiply the inequality by other positive number.
  Do you think the inequality will always remain
  true?

• 15 gt -6
• -?(15)gt-?(-6)
• -5 gt 2 false
• If you reverse the inequality, the statement is
  true.
• -5 lt 2 true
• Try other negative numbers as multipliers.

This suggests that when you multiply each side of
an inequality by a negative number, the order of
the inequality must be reversed. These examples
suggest the following properties.
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Multiplication/Division properties for
Inequalities
Multiplication and Division Properties of
Inequalities for positive numbers If a lt b and
c gt 0, then ac lt bc and    lt   If a gt b and
c gt 0, then ac gt bc and    gt
   Multiplication and Division Properties of
Inequalities for negative numbers If a lt b and
c lt 0, then ac gt bc and    gt   If a gt b and
c lt 0, then ac lt bc and    lt    Note All
the above properties apply to and .
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Example 2 Solve -0.5ylt6. Graph the solution
set.

• -0.5y lt 6
• (-2)(-0.5y) gt (-2)(6) Reverse the inequality
  sign because each side is multiplied by a
  negative
• y gt -12
• Any real number greater than -12 is a solution.

-12
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Example 3 Solve x x 4. 7Graph
the solution set.

• x x 4 7
• -7x x 4 Multiply each side by 7
• -8x 4 Add x to each side
• x - 1 Divide each side by -8 reversing the
  inequality sign.
• 2 A dot means this point is
  included.
• The solution set is xx -½.

-1/2
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Note

• Inequalities can be used to solve many verbal
  problems. You solve problems with inequalities
  the same way you solve problems with equations.

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Example 4 the problem

• Judy Kildow received a 10,000 inheritance that
  she wishes to invest. She wants to earn at least
  780 in interest this year so she can buy a
  stereo system with her earnings. She will invest
  some of the money in bonds that earn about 6
  interest annually and the rest in stock that she
  expects to earn 9 interest annually. What is
  the minimum she should invest in the stock? The
  phrase at least 780 means greater than or equal
  to 780

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Ex. 4

• EXPLORE Let n the amount invested in stocks.
  Then 10,000 n amount invested in bonds.
• PLAN
• (rate)(amount) (rate)(amount) min desired
• (0.09)(n) (0.06)(10,000 n) 780
• SOLVE
• 0.09n 600 0.06n 780 Set up equation
• 0.03n 780 Combine like terms
• n
  6000 Divide by 0.03 to solve
• Judy must invest at least 6,000 in stock.

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Dont forget the last step!

• EXAMINE OR CHECK Find the amount of interest
  she will earn from investing 6,000 in stocks and
  4,000 in bonds. Is the total at least 780?
• 6 of 4,000 240
• 9 of 6,000 540
• Total 780
• Check an amount more than 6,000 in stocks. Be
  sure the total is greater than 780.

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Finally . . .

• Set notation
• x x 5
• Dont forget to graph the solution set.
• Remind your teacher she has some number lines for
  you. They are generic and may not work in all
  settings.