3.1-3.2 Solving Inequalities

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3.1-3.2 Solving Inequalities

• Using Addition Subtraction

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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.
The number line is shaded to the left of 5, so
all numbers less than 5 are solutions
The open dot means that 5 is not a solution

• 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.
The number line is shaded to the right of -2, so
all numbers greater than -2 are solutions
The closed dot means that -2 is a solution

• 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
We will use the same steps that we did with
equations, if a number is added to the variable,
we add the opposite sign to both sides
w 5 (-5) lt 8 (-5)
w 0 lt 3
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 0 -10
w -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 0 gt 0
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 0 -3
y -3
All numbers from -3 down (including -3) make this
problem true!
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3.3 Solving Inequalities byMultiplying or
Dividing

• Objective Solve inequalities by using the
  multiplication and division properties of
  inequalities.

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5-Minute Check

• Solve each inequality.
• r 9 lt 4
• s 24 gt 23
• t (-30) 40
• u (-14) ³ -29
• 19 lt v 9 v gt 28

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3.3 Solving Inequalities byMultiplying or
Dividing
When you multiply or divide each side of an
inequality by a positive integer, the result
remains true.
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3.3 Solving Inequalities byMultiplying or
Dividing
Example 1 Solve 9x gt -36 and graph the solution
on a number line.
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3-7 Solving Inequalities byMultiplying or
Dividing
Example 1 Solve 9x gt -36 and graph the solution
on a number line. 9x gt -36
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3-3 Solving Inequalities byMultiplying or
Dividing
Example 1 Solve 9x gt -36 and graph the solution
on a number line. 9x gt -36
9x/9 gt -36/9 Divide each side by 9.
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3-3 Solving Inequalities byMultiplying or
Dividing
Example 1 Solve 9x gt -36 and graph the solution
on a number line. 9x gt -36
9x/9 gt -36/9 Divide each side by 9.
x gt -4
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3-3 Solving Inequalities byMultiplying or
Dividing
Example 1 Solve 9x gt -36 and graph the solution
on a number line. 9x gt -36
9x/9 gt -36/9 Divide each side by 9.
x gt -4
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Notice that -2 is included
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3-3 Solving Inequalities byMultiplying or
Dividing
Example 2 Solve -4x ³ 12 and graph the solution
on a number line.
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3-3 Solving Inequalities byMultiplying or
Dividing
Example 2 Solve -4x ³ 12 and graph the solution
on a number line. -4x ³ 12
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3-3 Solving Inequalities byMultiplying or
Dividing
Example 2 Solve -4x ³ 12 and graph the solution
on a number line. -4x ³ 12
-4x/(-4) 12/(-4) Divide each side by 4
and reverse the order symbol.
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3-3 Solving Inequalities byMultiplying or
Dividing
Example 2 Solve -4x ³ 12 and graph the solution
on a number line. -4x ³ 12
-4x/(-4) 12/(-4) Divide each side by 4 and
x -3 reverse the order symbol.
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3-3 Solving Inequalities byMultiplying or
Dividing
Example 2 Solve -4x ³ 12 and graph the solution
on a number line. -4x ³ 12
-4x/(-4) 12/(-4) Divide each side by 4 and
x -3 reverse the order symbol.
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Question?

• How is the inequality symbol related to the
  shading on the number line?
• When the inequality is written with variable on
  the left, the inequality symbol points in the
  direction of the shading.

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3-3 Solving Inequalities byMultiplying or
Dividing
Assignment do c, e, f in your class folder You
must solve, check your answer
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3.4 solve and Graph Multi-Step Inequalities
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Classwork

• Complete Solving Inequalities pages 1 -2 in your
  classwork folder.
• You must solve, check and graph