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6.5 Inequalities

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Bumper Cars You must be at least 130cm tall to ride the bumper cars. This can be represented by the inequality h 130.

In the Real World

Solving Inequalities6. 5

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The solution of an inequality is the set of numbers that you can substitute for the variable to make the inequality true.

An inequality is a mathematical sentence formed by placing an inequality symbol between two expressions.

Bumper Cars You must be at least 130cm tall to ride the bumper cars. This can be represented by the inequality h 130.

In the Real World

The graph of an inequality in one variable is the set of points on a number line that represents the solution of the inequality.

Solving Inequalities6 5.

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Graphing InequalitiesEXAMPLE 1

x < 2 All numbers less than 21–3 –2 –1 0 32–4 4

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Graphing InequalitiesEXAMPLE 1

x < 2 All numbers less than 2

x –1 All numbers less than or equal to –1

1–3 –2 –1 0 32–4 4

1–3 –2 –1 0 32–4 4

Solving Inequalities6 5.

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Graphing InequalitiesEXAMPLE 1

x < 2 All numbers less than 2

x –1 All numbers less than or equal to –1

x > 0 All numbers greater than 0

1–3 –2 –1 0 32–4 4

1–3 –2 –1 0 32–4 4

1–3 –2 –1 0 32–4 4

Solving Inequalities6 5.

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Graphing InequalitiesEXAMPLE 1

x < 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

1–3 –2 –1 0 32–4 4

1–3 –2 –1 0 32–4 4

1–3 –2 –1 0 32–4 4

1–3 –2 –1 0 32–4 4

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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 2 < 6Multiply each side by 2.

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Solving Inequalities

Solving an InequalityEXAMPLE 2

Solve d – 2 1.

d – 2 1

Add 2 to each side.

Write inequality.

Simplify.

d – 2 + 2 1 + 2

d 3

To graph d 3, use a closed dot and draw the arrow pointing to the left.

2–2 –1 0 1 43–3 5

Then graph the solution.

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Solving Inequalities

Solving an InequalityEXAMPLE 2

Solve d – 2 1.

d – 2 1

Add 2 to each side.

Write original inequality.

Simplify.

Write original equation.

Substitute 0 for d.

Solution

checks.

d – 2 + 2 1 + 2

d 3

To graph d 3, use a closed dot and draw the arrow pointing to the left.

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.

d – 2 1

–2 1

2–2 –1 0 1 43–3 5

Then graph the solution.

0 – 2 1?

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Solving Inequalities

Solving an InequalityEXAMPLE 3

Solve 5 + y <13

5 + y < 13

Subtract 5 to each side.

Write original inequality.

Simplify.

5 + y - 5 < 13 - 5

y < 8

To graph y < 8, use an open dot and draw the arrow pointing to the left.

73 4 5 6 982 10

Then graph the solution.

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Solving Inequalities

Solving an InequalityEXAMPLE 4

Solve 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 > -2 use a closed dot and draw the arrow pointing to the right.

Then graph the solution.

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2–2 –1 0 1 43–3 5

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Solving Inequalities

Solving an InequalityEXAMPLE 5

Solve 3 t > 12

3 t > 12

Divide each side by 3..

Write original inequality.

Simplify.

3 t > 12

t > 4

To graph t > 4 use an open dot and draw the arrow pointing to the right.

Then graph the solution.

6 5.

84 5 6 7 1093 11

3 3

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Solving Inequalities

Solving an InequalityEXAMPLE 6

Solve 3x + 5 < 23

3x + 5 < 23

Subtract 5 on each side

Write original inequality.

Simplify.

3x + 5 < 23

3x < 18

To graph x < 6 use an open dot and draw the arrow pointing to the left.

Then graph the solution.

6 5.

84 5 6 7 1093 11

-5 -5

x < 6 Divide both sides by 3 and simplify

Assignment: p. 248 9 - 22