Algebra II

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Algebra II

• Section 1-6
• Solving Inequalities

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What You'll LearnWhy It's Important

• To solve inequalities and graph the solution sets
• You can use inequalities to solve problems
  involving health, school and shopping

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Inequality Symbols

• less than lt
• (use an open circle when graphing)
• greater than gt
• (use an open circle when graphing)
• less than or equal to
• (use a closed circle when graphing)
• greater than or equal to
• (use a closed circle when graphing)

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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

• For any real numbers, a, b, and c
• 1. if a gt b, then a c gt b c and a c gt b c
• 2. if a lt b, then a c lt b c and a c lt b c
• Add or subtracting the same number to each side
  of an inequality does not change the truth of the
  inequality.

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Example 1

• Use the addition and subtraction properties for
  inequalities to solve this inequality, then graph
  the solution set on a number line.
• 6x 3 gt 5x - 2

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Example 1 Solution

• Solve 6x 3 gt 5x 2. Graph the solution set.
• 6x 3 gt 5x 2
• -5x -3 gt-5x 3
• x gt -5
• Any real number greater than -5 is a solution
• Check Substitute -5 for x in 6x 3 gt 5x 2.
  The two sides should be equal. Then substitute a
  number greater than -5. The inequality should be
  true.

An open circle means that this point is not
included
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Multiplication Division Properties for
Inequalities

• Does multiplying the same quantity on each side
  effect the inequality?
• -2 lt 10 now multiply by 3 on each side, is the
  inequality still a true statement?
• -2(3) lt 10(3)
• -6 lt 30
• What about multiplying by -3 on each side?
• -2(-3) lt 10(-3)
• 6 lt -30 (false statement)

• Does dividing by the same quantity on each side
  effect the inequality?
• 10 lt 50 now divide by 5 on each side, is the
  inequality still a true statement?
• -2 lt 10
• What about dividing by a -5?
• 2 lt -10 (false statement)

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Dont be a dummy!!

• When you multiply or divide an inequality by a
  negative number, it changes the direction of the
  inequality. 

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Multiplication Division Properties for
Inequalities

• For any real numbers a, b, and c
• 1. if c is positive and a lt b, then ac lt bc and
• 2. if c is positive and a gt b, then ac gt bc and
• 3. if c is negative and a lt b, then ac gt bc and
• 4. if c is negative and a gt b, then ac lt bc and

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Example 2

• Solve -0.4p gt 10. Graph the solution set.

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Example 2

• Solve -0.4p gt 10. Graph the solution set.
• p lt -25

Reverse the inequality because each side was
divided by a negative number
Any real number less than -25 is a solution
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Set Builder Notation

• The solution in Example 2 can be written using
  set-builder notation.
• This solution set can be written as pplt-25
• This is read as the set of all numbers p such
  that p is lessthan -25.

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Example 3

• Solve
• Write the answer in set builder notation
• Graph the solution set

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Example 3

• Solve

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Example 4Application School

• Ron's scores on the first three of four 100-point
  chemistry tests were 90, 96, and 86. What score
  must he receive on the fourth test to have an
  average of at least 92 for all the tests?

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Example 4 Application School

• Ron's scores on the first three of four 100-point
  chemistry tests were 90, 96, and 86. What score
  must he receive on the fourth test to have an
  average of at least 92 for all the tests?
• Explore Let x represent the score needed on the
  fourth test.
• The phrase at least 92 means greater than or
  equal to 92.
• Plan
• The average of Ron's test scores is their sum
  divided by 4.
• This number must be greater than or equal to 92.
• Write an inequality.
• Let x represent the score on the fourth test.
• Solve

• Examine Ron must score at least 96 on the
  fourth test to average at least 92 for all the
  tests.

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THE END