Algebra II

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Algebra II

• Section 1-7
• Solving Absolute Value Inequalities

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What You'll LearnWhy It's Important

• To solve compound inequalities using and and or,
  and
• To solve inequalities involving absolute value
  and graph the solutions
• You can use absolute value inequalities to solve
  problems involving entertainment and education

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Solve a Compound Inequality

• To solve a compound inequality, you must solve
  each part of the inequality.
• Thus, the graph of a compound inequality
  containing and is the intersection of the graphs
  of the two inequalities.
• The intersection can be found by graphing the two
  inequalities and then determining where these
  graphs overlap or intersect.

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Graph the solution set of x -2 and x lt 5
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Graph the solution set of x -2 and x lt 5
The solution set is x-2 x lt 5
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Example 1

• Solve 9 lt 3x 6 lt 15.
• Then graph the solution set.

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Example 1SolutionSolve 9 lt 3x 6 lt 15

• Method 1
• Write the compound inequality using the word and.
  Then solve each part.
• 9 lt 3x 6 and 3x 6 lt 15
• 3lt 3x 3x lt 9
• 1lt x and x lt 3

• Method 2
• Solve both parts at the same time by subtracting
  6 to each part of the inequality. Then divide
  each part by 3.
• 9 lt 3x 6 lt 15
• 9 6 lt 3x 6 6 lt 15 6
• 3 lt 3x lt 9
• 3 3 3
• 1 lt x lt 3

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Example 1 SolutionGraph the solution set of 1 lt
x and x lt 3
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Example 1 SolutionGraph the solution set of 1 lt
x and x lt 3
The solution set is x1lt x lt 3
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Solve a Compound Inequality

• Another type of compound inequality contains the
  word or instead of and.
• A compound inequality containing or is true if
  one or more of the inequalities is true.
• The graph of a compound inequality containing or
  is the union of the graphs of the two
  inequalities.

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Graph the solution set ofx -1 or x lt -4
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Graph the solution set ofx -1 or x lt -4
The last graph shows the solution set, xx -1
or x lt -4
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Example 2

• Solve x 3 gt 1 or x 2 lt 1
• Then graph the solution set.

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Example 2 Solution

• Solve x 3 gt 1 or x 2 lt 1
• x 3 3 gt 1 3 or x 2 2 lt 1 2
• x gt 4 or
  x lt -1

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Graph the solution set ofx gt 4 or x lt -1
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Graph the solution set ofx gt 4 or x lt -1
The last graph shows the solution set, xx gt 4
or x lt -1
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Absolute Value

• Recall the absolute value of a number is its
  distance from 0 on the number line.
• You can us this idea to solve inequalities
  involving absolute value

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x lt n

• x lt 4 means that the distance between 0 and x
  is less than 4 units
• Therefore, x gt -4 and x lt 4.
• The solution set is x-4 lt x lt 4
• So, if x lt n, then x gt -n and x lt n

4 units
4 units
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x gt n

• x gt 4, this means that the distance between 0
  and x is greater than 4 units
• Therefore, x lt -4 or x gt 4. The solution set is
  xx lt -4 or x gt 4
• So, if x gt n, then x lt -n or x gt n

4 units
4 units
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Example 3

• Solve y lt 7

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Example 3 Solution

• Solve y lt 7
• Means that the distance between y and 0 is less
  than 7 units.
• To make y lt 7 true, you must substitute values
  for y that are less than 7 units from 0.
• y lt 7 and y lt 7 y gt -7
• y lt 7 is the same as y lt 7 and y gt -7

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Graph the solution set of y lt 7 and y gt -7
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Graph the solution set of y lt 7 and y gt -7
The solution set is y-7 lt y lt 7
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Example 4

• Solve 2x 4 12.
• Graph the solution set.

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Example 4

• Solve 2x 4 12.
• The inequality says that 2x 4 is greater than
  or equal to 12 units from 0.
• 2x 4 12 or -2x - 4 12
• 2x 4 -4 12-4 -2x - 4 4 124
  2x 8 -2x 16 x 4
  or x -8

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Graph the solution set ofx 4 or x -8
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Graph the solution set ofx 4 or x -8
The last graph shows the solution set, xx 4
or x -8
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Example 5

• x 4lt 1 or x 2 gt 1

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Example 5

• x 4lt 1 or x 2 gt 1
• x lt 5 or x gt -1

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Graph the solution set of x lt 5 or x gt -1
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Graph the solution set of x lt 5 or x gt -1
The last graph shows the solution set, All Real
Numbers
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No Solution

• Some absolute value inequalities have no solution
• For example, 4x 3 lt -6 is never true
• Since the absolute value of a number is never
  negative, there is no replacement for x that will
  make this sentence true. So, The solution set to
  this inequality is the empty set. Ø

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Always True

• Other absolute value inequalities are always
  true. On such inequality is x 5 gt -10
• The solution set of this inequality is all real
  numbers. Can you see why?
• Think of the definition of absolute value!
• Because the absolute value of x 5 gt 0 and 0 gt
  -10

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THE END