Download a > -2

Inequalities
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7. 3-5
Solving Inequalities
In the Real World
Bumper Cars You must be at least 130cm tall to
ride the bumper cars. This can be represented
by the inequality h  130.
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7. 3-5 Solving Inequalities
In the Real World
Bumper Cars You must be at least 130cm tall to
ride the bumper cars. This can be represented
by the inequality h  130.
An inequality is a mathematical sentence formed
by placing an inequality symbol between two
expressions.
The solution of an inequality is the set of numbers that you can substitute for
the variable to make the inequality true.
The graph of an inequality in one variable is the set of points on a number
line that represents the solution of the inequality.
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Solving Inequalities
EXAMPLE
1
x<2
Graphing Inequalities
All numbers less than 2
–4 –3 –2 –1
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0
1
2
3
4
4
Solving Inequalities
EXAMPLE
1
x<2
x  –1
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Graphing Inequalities
All numbers less than 2
All numbers less
than or equal to –1
–4 –3 –2 –1
0
1
2
3
4
–4 –3 –2 –1
0
1
2
3
4
5
Solving Inequalities
EXAMPLE
1
x<2
x  –1
x>0
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Graphing Inequalities
All numbers less than 2
All numbers less
than or equal to –1
–4 –3 –2 –1
0
1
2
3
4
–4 –3 –2 –1
0
1
2
3
4
–4 –3 –2 –1
0
1
2
3
4
All numbers greater than 0
6
Solving Inequalities
EXAMPLE
1
x<2
x  –1
x>0
x  –2
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Graphing Inequalities
All numbers less than 2
All numbers less
than or equal to –1
–4 –3 –2 –1
0
1
2
3
4
–4 –3 –2 –1
0
1
2
3
4
–4 –3 –2 –1
0
1
2
3
4
–4 –3 –2 –1
0
1
2
3
4
All numbers greater than 0
All numbers greater
than or equal to –2
7
Solving Inequalities
Equivalent inequalities are 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 same positive number.
1<3
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Multiply each side by 2.
2<6
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Solving Inequalities
EXAMPLE
2
Solving an Inequality
Solve d – 2  1.
Then graph the solution.
d–21
Write inequality.
d–2+21+2
Add 2 to each side.
d3
Simplify.
To graph d  3, use a closed dot and
draw the arrow pointing to the left.
–3 –2 –1
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1
2
3
4
5
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Solving Inequalities
EXAMPLE
2
Solving an Inequality
Solve d – 2  1.
Then graph the solution.
d–21
Write original inequality.
d–2+21+2
d3
Add 2 to each side.
d–21
Simplify.
0–21
?
To graph d  3, use a closed dot and
draw the arrow pointing to the left.
–3 –2 –1
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1
CHECK To check the solution
d  3, choose any number less
than or equal to 3 to substitute for
d. The check below uses d = 0.
2
3
4
–2  1
Write original equation.
Substitute 0 for d.
Solution checks.
5
10
Solving Inequalities
EXAMPLE
3
Solving an Inequality
Solve 5 + y <13
Then graph the solution.
5 + y < 13
Write original inequality.
5 + y - 5 < 13 - 5
Subtract 5 to each side.
y<8
Simplify.
To graph y < 8, use an open dot and
draw the arrow pointing to the left.
2
3
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5
6
7
8
9 10
11
Solving Inequalities
EXAMPLE
4
Solving an Inequality
Solve a – 3 > -5
Then graph the solution.
a – 3 > -5
Write original inequality.
a – 3 + 3 > -5 + 3
Add 3 to each side..
a > -2
Simplify.
To graph a > -2 use a closed dot and
draw the arrow pointing to the right.
–3 –2 –1
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1
2
3
4
5
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Solving Inequalities
EXAMPLE
5
Solving an Inequality
Solve 3 t > 12
Then graph the solution.
3 t > 12
Write original inequality.
3 t > 12
3
3
t>4
Divide each side by 3..
Simplify.
To graph t > 4 use an open dot and
draw the arrow pointing to the right.
3
4
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6
7
8
9 10 11
13
Solving Inequalities
EXAMPLE
6
Solving an Inequality
Solve 3x + 5 < 23
Then graph the solution.
3x + 5 < 23
Write original inequality.
3x + 5 -5 < 23 -5
Subtract 5 on each side
3x < 18
Simplify.
Divide both sides by 3 and simplify
x<6
To graph x < 6 use an open dot and
draw the arrow pointing to the left.
3
4
5
6
7
8
9 10 11
Assignment: p. 248 9 - 22
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