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Inequalities

Mathematical sentences that use any of the following symbols

|> Greater than |< Less than |≤ Less than or equal to |≥ Greater than or equal to |

Solving Inequalities

• Done the same way you solve equations.

• Exception: when you multiply or divide both sides of an inequality by a negative number, you must change the direction of the inequality symbol.

Example: Solving Inequalities Using Addition/Subtraction

Solve the following inequalities and graph the solution on the number line.

a. y + 3 > 5 b. x - 3 < 5

Step 1: Isolate y variable Step 1: Isolate the x variable

Subtract 3 from both sides add 3 to both sides

y + 3 > 5 x - 3 < 5

- 3 > -3 + 3 < +3

y > 2 x < 9

[pic] [pic]

Example: Solving Inequalities Using Multiplication/Division

Solve the following inequalities and graph the solution on the number line.

a. 4y > 12 b. –3y > 15

Step 1: Isolate y variable Step 1: Isolate the y variable

divide both sides by 3 divide both sides by -3

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y > 3 y < -5 (since we divided by negative, ineq switched)

[pic] [pic]

Solving Inequalities (Multi-Step)

• Complete the Distributive Property

• Simplify by adding like terms.

• Eliminate the variable on 1 side

• Eliminate constant term on the side with variable

• Solve for the variable

• Check solution

• Remember: addition/subtraction must be done before multiplication/division

• Note: some inequalities have no solution and others are true for all real numbers.

Example: Solving Inequalities (Multi-Step)

Solve the following inequalities and graph the solution on the number line.

a. 2y + 3 < 9 b. 3y + 2y > 15

Step 1: Opposite of add is subtract Step 1: Add like terms

So subtract 3 from both sides So add 3y + 2y

Step 2: Perform the necessary operation Step 2: Perform the necessary operation

2y + 3 < 9 5y > 15

- 3 -3

2y < 6

Step 3: Opposite of multiply is divide Step 3: Opposite of multiply is divide

So, divide by 2 So, divide by 5

Step 4: Perform the necessary operation Step 4: Perform the necessary operation

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y < 3 y > 3

[pic]

Compound Inequalities

• 2 inequalities joined by the word “and” or “or”

• Example: -5≤ x ≤ 7 is the same as x ≥ -5 and x ≤ 7

Example: Solving Compound Inequalities (and/or)

Solve the following compound inequalities and graph the solution on the number line.

a. –4 < r –5 ≤ -1 b. 4v + 3 < -5 or –2v + 7 < 1

Step 1: Isolate the variable r Step 1: Isolate the variable v

Add 5 to all sides 4v + 3 < -5 or –2v + 7 < 1

–4 < r –5 ≤ -1 -3 -3 -7 -7

+5 +5 +5 [pic]or [pic]`

1 < r ≤ 4 v < -2 or v > 3

Absolute Value Equations & Inequalities

• Since absolute value represents distance, it can never be negative

• When solving for |a| = b, 2 solutions a = b and a = -b

• When solving for |a| < b, solving for –b < a < b

• When solving for |a| > b, solving for a < -b or a > b

Example: Solving Absolute Value Equations

Solve the following equations.

a. | x | + 5 = 11 b. |2p + 5| =11

Step 1: Isolate absolute value function

|x | + 5 – 5 = 11 – 5 |2p + 5| = 11

Step 2: Simplify

|x| = 6

Step 3: Write 2 equations & solve

x = 6 or x = -6 2p + 5 = 11 or 2p + 5 = -11

-5 = -5 -5 = -5

2p = 6 or 2p = -16

p = 3 or p = -8

Example: Solving Absolute Value Inequalities

Solve the following inequalities. Check and graph your solution.

a. |n -1 | < 5 b. |v -3| ≥ 4

n – 1 < 5 or n –1 > -5 v – 3 ≥ 4 or v – 3 ≤ -4

n –1 + 1 < 5 + 1 or n –1 + 1 > -5 +1 v – 3 + 3 ≥ 4 + 3 or v –3 + 3 ≤ -4 + 3

n < 6 or n > -4 v ≥ 7 or v ≤ -1

Representing Inequalities Practice

Inequality Interval Graph Set Notation

1. 5< x < 8

2. x < -3

3. x > 0

Rewrite as an inequality and graph:

4. [5, 9) 5. [-3, 10]

6. (-(, 5] 7. (-3, ()

Rewrite in interval notation, set notation, and graph:

8. -6 < x < 6 9. x > -3

10. x < -2 11. -2 < x < 5

12. x < 4 or x > 6 13. x < -2 or x > 0

Write as an inequality, in interval notation and in set notation:

14. 15.

16. 17.

18. 19.

Practice: Solving Inequalities Using Addition/Subtraction and Multiplication/ Division

Solve the following inequalities. Graph your solution.

1. x – 3 < 5 2. 12 ≤ x – 5

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3. n – 7 ≤ -2 4. –4 > b - 1

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5. [pic]n ≤ 2 6. 6 ≤[pic]w

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7. [pic] < -1 8. –20 > -5c

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Practice: Solving Inequalities (Multi-Step)

Solve the following inequalities and graph the solution on the number line.

9. -15c – 28 > 152 10. 4x – x + 8 ≤ 35

[pic]

11. 2x – 3 > 2(x-5) 12. 7x + 6 ≤ 7(x – 4)

[pic]

Practice: Solving Compound Inequalities (and/or)

Solve the following compound inequalities and graph the solution on the number line.

13. –6 < 3x < 15 14. –3 < 2x – 1 < 7

[pic]

15. 7 < -3n + 1 ≤ 13 16. –2x + 7 > 3 or 3x – 4 ≥ 5

[pic]

17. 2d + 5 ≤ -1 or –2d + 5 ≤ 5 18. 3x + 2 < -7 or –4x + 5 < 1

[pic]

Practice: Solving Absolute Value Equations

Solve the following equations. Check your solution.

19. |t| -2 = -1 20. 3|n| = 15 21. 4 = 3|w| - 2

Practice: Solving Absolute Value Inequalities

Solve the following inequalities. Graph your solution.

22. |w + 2 | > 5 23. |y – 5| ≤ 2

24. [pic] 25. [pic]

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-8

-1 3

3

-4

7

18

-9

6

44

45

How to solve Algebra 1 equations with variables on both sides? Algebra 1: 3.4 SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES Goal: Get ONE variable alone on one side of = sign. Use Distributive Property, if necessary. Combine like terms, if necessary