Solving Multi-Step Equations and Inequalities

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Solving Multi-Step Equations and Inequalities
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Solving Equations with variables on both sides

• You will need to add or subtract to get the
  variables on one side and the constant term on
  the other side.

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5x 3 2x 12

• 5x 3 2x 12
• 5x (2x) 3 2x (-2x) 12
• 3x 3 12
• 3x 3(3) 12(3)
• 3x 15
• (3x)/3 15/3
• x 5

• given
• / - prop
• simplified
• / - prop
• simplified
• ? / ? prop
• ?

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Solving Inequalities

• x 3 ? 5
• x 3 3 ? 5 3
• x ? 8
• 8
• When the letter is on the left, you shade the way
  the arrow is pointing.

• given
• /- prop of order
• ? letter on left
• Graphed
• When the inequality is equal to the graph is a
  solid dot

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-5 gt b 1

• -5 gt b 1
• -51 gt b 11
• -4 gt b
• -4
• When the letter is on the right, you shade the
  opposite way the arrow is pointing.

• given
• / - prop of order
• ?
• graphed
• When the inequality is not equal to the dot is
  open.

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Multiplying and Dividing to solve an inequality
i.e., 2/3 x ? 8

• 2/3 x ? 8
• (3/2) 2/3 x ? 8(3/2)
• x ?12
• 12

• given
• ? / ? prop
• ?
• Graphed
• Stop and think, did I multiply or divide by a
  negative? If I did, the inequality symbol flips
  the opposite way.

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-3x ? 15

• -3x ? 15
• -3x/-3 ? 15/-3
• x ? -5
• -5

• given
• ? / ? prop of order
• ?
• graphed
• Stop and think, did I multiply or divide by a
  negative? If I did, the inequality symbol flips
  the opposite way.

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Combined Inequalities

• Inequalities can be combined
• We can look at a union of inequalities
• We can look at an intersection of inequalities

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Intersection

• Only use the parts that both inequalities have

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Union

• Put together everything from two inequalities

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Solving Absolute Value Equations

• An absolute value equation is one that has the
  variable inside the absolute value marks.
• ?x? 3
• The value of x could be 3 or -3

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?x? 5 11

• ?x? 5 11
• ?x? 5 -5 11 -5
• ?x? 6
• x 6 and x -6

• given
• / - prop
• simplified
• ?

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?p - 9? 3

• ? p - 9? 3
• p-93 and p-9 -3
• p-9939 p-99 -39
• p 12 p6

• Given
• The absolute value could be 3 or -3
• / - prop
• ?

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Solving absolute value inequalities

• means x is 3 units from 0
• means x is less than 3 units from 0
• means x is more than 3 units from 0

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Solve x lt 3 (conjunction)

• This means x is all the real numbers less than 3
  units from 0 on the number line.

0
3
-3
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Solve x gt 3 (disjunction)

• This means x is all the real numbers greater than
  3 units from 0 on the number line.

0
3
-3
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Compare the two graphs
x lt 3 is a conjunction
3
-3
x gt 3 is a disjunction
3
-3
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Solve this absolute value inequality

• x gt 5
• This means that the answer is less than -5 or
  greater than 5 (disjunction)
• xlt -5 x gt 5

-5
5
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