Solving and Graphing Inequalities

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Solving and Graphing Inequalities
CHAPTER 6 REVIEW
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An inequality is like an equation, but instead of
an equal sign () it has one of these signs lt
less than less than or equal to gt
greater than greater than or equal to
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x lt 5

• means that whatever value x has, it must be less
  than 5.
• Try to name ten numbers that are less than 5!

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Numbers less than 5 are to the left of 5 on the
number line.

• If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you
  are right.
• There are also numbers in between the integers,
  like 2.5, 1/2, -7.9, etc.
• The number 5 would not be a correct answer,
  though, because 5 is not less than 5.

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x -2

• means that whatever value x has, it must be
  greater than or equal to -2.
• Try to name ten numbers that are greater than or
  equal to -2!

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Numbers greater than -2 are to the right of 5 on
the number line.

• If you said -1, 0, 1, 2, 3, 4, 5, etc., you are
  right.
• There are also numbers in between the integers,
  like -1/2, 0.2, 3.1, 5.5, etc.
• The number -2 would also be a correct answer,
  because of the phrase, or equal to.

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Where is -1.5 on the number line? Is it greater
or less than -2?
-2

• -1.5 is between -1 and -2.
• -1 is to the right of -2.
• So -1.5 is also to the right of -2.

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Solve an Inequality
w 5 lt 8
w 5 (-5) lt 8 (-5)
All numbers less than 3 are solutions to this
problem!
w lt 3
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More Examples
8 r -2
8 r (-8) -2 (-8)
r -10
All numbers from -10 and up (including -10) make
this problem true!
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More Examples
x - 2 gt -2
x (-2) (2) gt -2 (2)
x gt 0
All numbers greater than 0 make this problem true!
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More Examples
4 y 1
4 y (-4) 1 (-4)
y -3
All numbers from -3 down (including -3) make this
problem true!
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There is one special case.

• Sometimes you may have to reverse the direction
  of the inequality sign!!
• That only happens when you
• multiply or divide both sides of the
  inequality by a negative number.

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Example

• Solve -3y 5 gt23
• -5 -5
• -3y gt 18
• -3 -3
• y lt -6

• Subtract 5 from each side.
• Divide each side by negative 3.
• Reverse the inequality sign.
• Graph the solution.

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Try these

• 1.) Solve 2x 3 gt x 5 2.)Solve - c
  11 gt23
• 3.) Solve 3(r - 2) lt 2r 4

-x -x x 3 gt 5 -3 -3 x
gt 2
11 11 -c gt 34 -1 -1 c lt -34
3r 6 lt 2r 4 -2r -2r r 6 lt 4 6
6 r lt 10
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You did remember to reverse the signs . . .
. . . didnt you?
Good job!
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- 4x - 4x
Ring the alarm! We divided by a negative!
6 6
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Remember Absolute Value
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Ex Solve 6x-3 15

• 6x-3 15 or 6x-3 -15
• 6x 18 or 6x -12
• x 3 or x -2
• Plug in answers to check your solutions!

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Ex Solve 2x 7 -3 8

• Get the abs. value part by itself first!
• 2x7 11
• Now split into 2 parts.
• 2x7 11 or 2x7 -11
• 2x 4 or 2x -18
• x 2 or x -9
• Check the solutions.

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Ex Solve graph.

• Becomes an and problem

-3 7 8
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Solve graph.

• Get absolute value by itself first.
• Becomes an or problem

-2 3 4
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Example 1
This is an or statement. (Greator).
Rewrite. In the 2nd inequality, reverse the
inequality sign and negate the right side
value. Solve each inequality. Graph the solution.

• 2x 1 gt 7
• 2x 1 gt 7 or 2x 1 gt7
• 2x 1 gt7 or 2x 1 lt-7
• x gt 3 or x lt -4

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Example 2
This is an and statement. (Less thand).
Rewrite. In the 2nd inequality, reverse the
inequality sign and negate the right side
value. Solve each inequality. Graph the
solution.

• x -5lt 3
• x -5lt 3 and x -5lt 3
• x -5lt 3 and x -5gt -3
• x lt 8 and x gt 2
• 2 lt x lt 8

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Absolute Value Inequalities
Case 1 Example
and
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Absolute Value Inequalities
Case 2 Example
or
OR
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Absolute Value

• Answer is always positive
• Therefore the following examples
• cannot happen. . .
• Solutions No solution

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Some Helpful Hints

• If the sign is gt or lt the line is dashed
• If the sign is ? or ? the line will be solid
• When dealing with just x and y.
• If the sign gt or ? the shading either goes up
  or to the right
• If the sign is lt or ? the shading either goes
  down or to the left

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Graphing an Inequality in Two Variables
Graph x lt 2










Step 1 Start by graphing the line x 2
Now what points would give you less than 2?
Since it has to be x lt 2 we shade everything to
the left of the line.
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Graphing a Linear Inequality
Sketch a graph of y ? 3










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Using What We Know
Sketch a graph of x y lt 3










Step 1 Put into slope intercept form y lt-x 3
Step 2 Graph the line y -x 3