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6.5 Inequalities
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Bumper Cars You must be at least 130cmtall to
ride the bumper cars. This can be represented
by the inequality h130.
In the Real World
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The solution of an inequalityis the set of numbers that you can substitute forthe variable to make the inequality true.
An inequalityis a mathematical sentence formed
by placing an inequality symbol between two
expressions.
Bumper Cars You must be at least 130cmtall to
ride the bumper cars. This can be represented
by the inequality h130.
In the Real World
The graph of an inequalityin one variable is the set of points on a number
line that represents the solution of the inequality.
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Graphing Inequalities
EXAMPLE 1x< 2 All numbers less than 2
13 2 1 0 324 4
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Graphing Inequalities
EXAMPLE 1x< 2 All numbers less than 2
x1 All numbers less
than or equal to
1
13 2 1 0 324 4
1
3
2
1 0 32
4 4
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Graphing Inequalities
EXAMPLE 1x< 2 All numbers less than 2
x 1 All numbers less
than or equal to
1
x> 0 All numbers greater than 0
13 2 1 0 324 4
1
3
2
1 0 32
4 4
13 2 1 0 324 4
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Graphing Inequalities
EXAMPLE 1x< 2 All numbers less than 2
x 1 All numbers less
than or equal to
1
x> 0 All numbers greater than 0
x 2 All numbers greater
than or equal to
2
13 2 1 0 324 4
1
3
2
1 0 32
4 4
13 2 1 0 324 4
1
3
2
1 0 32
4 4
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Solving Inequalities
Equivalent inequalitiesare inequalities that have the same solution. You
can produce an equivalent inequality in the following ways:
Add or subtract the same number on each side.
Multiply or divide each side by the samepositivenumber.
1 < 3 2 < 6Multiply each side by 2.
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Solving Inequalities
Solving an Inequality
EXAMPLE 2Solve d2 1.
d2 1
Add 2to each side.
Write inequality.
Simplify.
d2 + 2 1 + 2
d 3
To graph d3,use a closed dot and
draw the arrow pointing to the left.
22 1 0 1 433 5
Then graph the solution.
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Solving Inequalities
Solving an Inequality
EXAMPLE 2Solve d2 1.
d2 1
Add 2to each side.
Write original inequality.
Simplify.
Write original equation.
Substitute 0for d.
Solution checks.
d2 + 2 1 +2
d3
To graph d3,use a closed dot and
draw the arrow pointing to the left.
CHECK To check the solution
d3,choose any number less
than or equal to 3to substitute for
d. The check below uses d= 0.
d2 1
2 1
22 1 0 1 433 5
Then graph the solution.
02 1?
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Solving Inequalities
Solving an Inequality
EXAMPLE 4Solve a 3 > -5
a 3 > -5
Add 3 to each side..
Write original inequality.
Simplify.
a 3 3> -5 3
a > -2
To graph a > -2use a closed dot anddraw the arrow pointing to the right.
Then graph the solution.
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22 1 0 1 433 5
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Solving Inequalities
Solving an Inequality
EXAMPLE 5Solve 3 t > 12
3 t > 12
Divide each side by 3..
Write original inequality.
Simplify.
3 t > 12
t > 4
To graph t > 4use an open dot anddraw the arrow pointing to the right.
Then graph the solution.
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3 3
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Solving Inequalities
Solving an Inequality
EXAMPLE 6Solve 3x 5 < 23
3x 5 < 23
Subtract 5 on each side
Write original inequality.
Simplify.
3x 5 < 23
3x < 18
To graph x < 6use an open dot anddraw the arrow pointing to the left.
Then graph the solution.
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84 5 6 7 1093 11
5 5
x < 6 Divide both sides by3 and simplify
Assignment: p. 248 9 - 22