Algebra: Chapter 3 Notes

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Notes #16: Sections 3-1, 3-2, and 3-6 Section 3-1: Inequalities and their graphs A. Identifying Solutions of Inequalities A solution of an inequality is any number that makes the inequality _______. For example, the solutions of the inequality x < 3 are all: _________________ Identifying Solutions by Mental Math Is each number a solution of x 7? Meaning, does the value of x make the inequality ______? 1.) 9 Solution: ________

2.) 14

2

Solution: ________

3.) -5 Solution: ________

Identifying Solutions by Evaluating Is each number a solution of 6x - 3 > 10? Meaning, does the value of x make the inequality ______? 4.) 3 Solution: ________

5.) 4 Solution: ________

B. Graphing and Writing Inequalities in One Variable How many solutions are there to an inequality like m < -3.5? _______________ Rather than list solutions, you can use a graph to indicate all of the solutions of an inequality. When graphing an inequality on a number line, follow these tips:

Always arrange the final inequality so the variable is on the _____________ side Label your number so that the number in the solution is in the ___________ of the graph Use a __________________ for < or > and use a ___________________ for ≤ or ≥ Graph to the RIGHT when a ______ or ______ and to the LEFT when a _______ or ______

6.) x < 3

7.) m -2

8.) c > -2

9.) -4 p

Algebra: Chapter 3 Notes

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Write an inequality for each graph. Variable choice may vary. 10.)

11.)

12)

13)

Section 3-2: Solving Inequalities Using Addition and Subtraction A. Using Addition and Subtraction to Solve Inequalities

**When solving an inequality, you can add and subtract the same number from

_______ _________ without changing the inequality sign** Solve the following inequalities, then graph the solutions. 14.) x – 3 < 5.

15.) m – 6 > -4

16.) y + 5 < -7 17.) 4.5 2.3g

Algebra: Chapter 3 Notes

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18.) 3.8 d + 7 19.) 12 x – 5

20.) 4

75

c 21.) 2 4

4 13 5

x

Example (Application) 22.) In order to receive a B in your literature class, you must earn more than 350 points of reading credits. Last week you earned 120 points. This week you earned 90 points. How many more points must you earn to receive a B?

Algebra: Chapter 3 Notes

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Section 3-6: Absolute Value Equations and Inequalities A. Solving Absolute Value Equations What does absolute value mean? Recall that the absolute value of a number is its ____________ from zero on a number line. Since absolute value represents _______________, it can never be ___________________. What does solving an absolute value equation mean?

3x means to find the places on the number line that are _________ away from _______.

Solution: ________________

3x means to find the places on the number line that are _________ away from _______.

Solution: ________________ What does the graph of an absolute value equation look like? The graph of 3x is below:

Solving Absolute Value Equations:

Get the | | alone Write two equations; one ___________ and one _____________ Solve for x; expect __________ answers Check both answers by _____________________________

Solve each equation. Check your solution. 1.) 5 11x

(check)

2.) 3 15n

(check)

3.) 3 2 4w

(check)

Algebra: Chapter 3 Notes

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4.) 3 5m

(check)

5.) 3 5 6r

(check)

6.) 2 3 12p

(check)

Algebra: Chapter 3 Notes

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Notes #17: Sections 3-6, 3-3 and 3-4 1.) 2 5 11p

(check)

2.) 2 3 5 1 9x

(check)

3.) 3 4 2 6x

(check)

Algebra: Chapter 3 Notes

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Section 3-3: Solving Inequalities Using Multiplication and Division

When multiplying and dividing the same number to both sides of an inequality,

follow these rules: If you multiply or divide by a positive number, leave the inequality sign ______ ________

If you multiply or divide by a negative number, _______ the inequality sign.

Explore why: Solve and graph the solution:

1.) 12

x

2.) 2 8t

3.) 0.6 0.2n

4.) 14

k

5.) 4 24c

6.) 3

65

w

7.) 5 25z

8.) 1

2t

9.) 4

83

y

Algebra: Chapter 3 Notes

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C. Application 13.) Your family budgets $160 to spend on fuel for a trip. How many times can they fill the car’s gas tank if it costs $25 each time?

Section 3-4: Solving Multi-Step Inequalities D. Solving inequalities with variables on one side Sometimes you need to perform two or more steps to solve an inequality. Your goal is still the same: to _______________ the variable on the ________ side of the inequality sign. Solve and graph your solution. 14.) 5 + 4b < 21

15.) 2 – 8x > -6

16.) 8z – 6 < 3z + 12

17.) 6z – 15 < 4z + 11

10.) 1

4 2

b

11.) 3

35

x

12.) 23

z

Algebra: Chapter 3 Notes

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18.) 3x + 4(6 – x) < 2

19.) 5(-3 + d) 3(3d – 2)

20.) 4 3 1

25 10 2

m m

21.) 5 1

2( 3) 16 9

y y

Algebra: Chapter 3 Notes

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Notes #18: Section 3-5 Section 3-5: Compound Inequalities Two inequalities that are joined by the word ____________ or the word ____________ form a ____________________________. Examples: A. Solving Compound Inequalities Containing AND What does it mean? A solution of an “and” compound inequality is any number that makes __________ inequalities true. Example: Find a solution for the following inequality x < 9 and x > 7 _____________ How do I write it? You can write an “AND” compound inequality as a sANDwich You can write the compound inequality x -5 and x 7 as: ____________________________. How do I say it? There are two correct ways to say this:

1) x is ___________________ -5 and ___________________ to 7. 2) x is ___________ -5 and 7 _________________.

How do I graph it? The solution of this inequality can be expressed with the following graph: Write a compound inequality that represents each situation. Graph the solution. 1.) All real numbers that are at least -2 and at most 4.

2.) All real numbers greater than -2 but less than 9.

3.) The books were priced between $3.50 and $6.00, inclusive.

Solve the inequality. Graph the solution. 4.) Solve -4 < r - 5 -1

5.) -6 3x < 15 6.) -3 < 2x – 1 13

Algebra: Chapter 3 Notes

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B. Solving Compound Inequalities joined by an OR What does it mean? A solution of an “or” compound inequality is any number that makes __________ inequality true. Example: Find a few solutions for the following inequality x > 3 or x < -2 _____, _____, _____ How do I write it? You cannot write an “or” compound inequality as one equation. You must write the solution as _____ inequalities separated by an _____. How do I say it? There is one correct way to say this:

x is ___________________ 3 or ___________________ -2. How do I graph it? The solution of this inequality can be expressed with the following graph: Write a compound inequality that represents each situation. Graph the solution. 7.) All real numbers that are less than 0 or greater than 3.

8.) Discounted tickets are available to children under 7 years old or to adults 65 and older.

Solving a Compound Inequality Containing Or 9.) Solve the compound inequality 3x+ 2 < -7 or -4x + 5 < 1

10.) Solve the compound inequality 4v + 3 < -1 or -2v + 7 < 1

Algebra: Chapter 3 Notes

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C. Application 11.) Your test grades in science so far are 83 and 87. What possible grades can you make on your next test to have an average between 85 and 90, inclusive?

Algebra: Chapter 3 Notes

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Notes#19: Section 3-6 Section 3-6: Solving Absolute Value Inequalities A. Solving Absolute Value Inequalities What do absolute value inequalities mean?

2x means “What numbers are _______________________ 2 units away from zero?” Graph the

solution: Is this an “AND” or an “OR” graph?

2y means “What numbers are _______________________ 2 units away from zero?” Graph the

solution: Is this an “AND” or an “OR” graph? Solving Absolute Value Inequalities:

Get | | alone Write 2 equations, one __________ and one ________________ (SWITCH THE SIGN!)

If , use ______________

If , , use _______________

Graph and solve for x (sometimes, put back in sandwich) Solve the absolute value inequalities. Graph your solutions. 1.) 5 14d 2.) 3 8k

Algebra: Chapter 3 Notes

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3.) 3 4v

4.) 5 2y

5.) 4 3 1x

6.) 6 2c

7.) 2 3 8y

8.) 4 3 3m

9.) 2 3 2 4x

10.) 5 2 3y