Solving Linear Inequalities Chapter 1.6 Part 3. Properties of Inequality 2

Solving Linear Inequalities

Chapter 1.6 Part 3

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Properties of Inequality

• The properties of inequality are very similar to the properties of equality, but with one important exception

• Consider the integers and

• The inequality is true since is located to the left of

• Suppose that we multiply each side of this inequality by

• The numbers then become and

• But now, the inequality is false

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Properties of Inequality

• We can make the inequality true by changing the inequality sign

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Properties of Inequality

• We can use the following properties to solve inequalities

• If a, b, and c are real numbers with , then

Property Meaning

You can add the same value to both sides

You can subtract the same value from both sides

You can multiply both sides by the same positive value

You can divide both sides by the same positive value

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Properties of Inequality

• These last two properties are special to inequalities

• If a, b, and c are real numbers with , then

• The same properties apply for

Property Meaning

You can multiply each side by the same negative number by changing the inequality sign

You can divide both sides by the same negative number by changing the inequality sign

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Example: Solve an Inequality with Variable Terms on Each SideSolve , then graph the solution.

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Example: Solve an Inequality with Variable Terms on Each Side

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Example: Solve an Inequality with Variable Terms on Each SideIf we pick a value in the shaded area (for example, 0) we end up with a true statement:

If we pick a value to the right of 3 (for example, 4) we get a false statement:

The statement is false because 4 is not a solution.

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Guided Practice

Solve each inequality, then graph the solution.

a)

b)

c)

d)

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Guided Practice

Solve each inequality, then graph the solution.

a)

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Guided Practice

Solve each inequality, then graph the solution.

b)

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Guided Practice

Solve each inequality, then graph the solution.

c)

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Guided Practice

Solve each inequality, then graph the solution.

d)

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Solving Compound Inequalities

• Remember that compound inequalities are actually made of two inequalities

• Solving compound inequalities amounts to solving each part

• In the case of a conjunction in the form , you will simply add/subtract or multiply/divide all three parts

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Example: Solving a Disjunction Inequality

Solve Solve each part separately and keep “or” in the final answer.

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Example: Solving a Disjunction Inequality

Solve

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Example: Solve a Conjunction Inequality

Solve .Remember that you want the x alone between the two inequality signs. Begin by adding 10 three times…

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Example: Solve a Conjunction Inequality

Solve .

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Guided Practice

Solve each inequality, then graph the solution.

a)

b)

c)

d)

e) A monitor lizard has a temperature that ranges from 18˚C to 34˚C. Write the range of temperatures as a compound inequality, then write an inequality giving the temperature range in Fahrenheit.

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Guided Practice

Solve each inequality, then graph the solution.

a)

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Guided Practice

Solve each inequality, then graph the solution.

b)

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Guided Practice

Solve each inequality, then graph the solution.

c)

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Guided Practice

Solve each inequality, then graph the solution.

d)

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Guided Practice

Solve each inequality, then graph the solution.

e) A monitor lizard has a temperature that ranges from 18˚C to 34˚C. Write the range of temperatures as a compound inequality, then write an inequality giving the temperature range in Fahrenheit.

Since the range is between 18 and 34, using as the variable we have

Replace by because these are equal:

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Guided Practice

Solve each inequality, then graph the solution.

e) A monitor lizard has a temperature that ranges from 18˚C to 34˚C. Write the range of temperatures as a compound inequality, then write an inequality giving the temperature range in Fahrenheit.

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Exercise 1.7

• Handout