(a) Solving Inequalities Examples; (b) - Beacon Learning Center

Solving Inequalities Examples 

1. Joe and Katie are dancers. Suppose you compare their weights. 

You can make only one of the following statements. 

Joe’s weight is 

less than Kate’s 

weight. 


the same as 

Kate’s weight. 


greater than Kate’s 

weight. 

2. Let j stand for Joe’s weight and k stand for Katie’s weight. Then 

you can use inequalities and an equation to compare their weights. 

j < k J = k j > k 

This is an illustration of the trichotomy property 

3. Trichotomy Property – For any two real numbers a and b, exactly one of the 

following statements is true. 

a b 

4. 

Addition and Subtraction 

Properties for Inequalities 

1. If a > b, then a + c > b + c and a – c > b – c. 

2. If a < b, then a + c 

Adding the same number to each side 

of an inequality does not change the 

truth of the inequality. 

5. The following numerical examples may be used to illustrate the Addition and 

Subtraction Properties. 

5 < 6 

5 + 2 < 6 + 2 

7 < 8 

–6 > –10 

–6 – 9 > –10 – 9 

–15 > –19 

These properties can be used to 

solve inequalities. Each solution 

set can be graphed on the 

number line. 

Solving Inequalities Johnny Wolfe www.BeaconLC.org Jay High School Santa Rosa County Florida September 22, 2001

6. Example – Solve 9x + 7 < 8x – 2. Graph the solution set. 

9x + 7 < 8x – 2 

–8x + 9x + 7 < –8x + 8x – 2 

x + 7 < –2 

x + 7 + (–7) < –2 + (–7) 

x < –9 

To check inequalities, first check the boundary point for the variable, and see 

if the two sides are equivalent. A true equation should occur if the inequality 

sign is replaced by the equals sign. To make sure the direction of the 

inequality is correct, check a point on each side of the boundary point. 

Check 

9(–9) + 7 = 8(–9) – 2 

–74 = –74 

Choose a point in the solution set. 

9(–10) + 7 < 8(–10) 

–87 < -80 TRUE 

Choose a point outside the 

solution set. 

9(–8) + 7 < 8(–8) 

–65 < –64 FALSE 

7. Example – Solve y + 6 > 3. Graph the solution set. 

y + 6 > 3 

y + 6 – 6 > 3 – 6 

y > –3 

8. Example – Solve –18 < t – 7. Graph the solution set. 

–18 < t – 7 

–18 + (7) < t – 7 + (7) 

–11 < t 

9. Example – Solve 2m + 9 < m + 4. Graph the solution set. 

2m + 9 < m + 4 

m + 9 < 4 

m < – 5 

10. We know that 18 > –11 is a true inequality. If you multiply each side of this 

inequality by a positive number, the result is a true inequality. 

18 > –11 

18 (3) > –11(3) 

54 > –33 TRUE 

What happens if we multiply by a 

negative number? 


11. Suppose you multiply each side of a true inequality by a negative number. 

18 > –11 

18(–2) > –11(–2) 

–36 > 22 FALSE! 

We must reverse the inequality 

symbol when we multiply or 

divide by a negative number. 

12. 

Multiplication and Division Properties for Inequalities 

1. If c is positive and a < b, then ac < bc and c 

a < c 

b . 

2. If c is positive and a > b, then ac > bc and c 

a > c 

b . 

3. If c is negative and a < b, then ac > bc and c 

a > c 

b . 

4. If c is negative and a > b, then ac > bc and c 

a < c 

b . 

13. The following examples may be used to illustrate the Multiplication and Division 

Properties. 

16 > –8 

–5 < –2 

16 − 8 

(–5)( –4) > –2(–4) 

< 

− 2 − 2 

20 > 8 

–8 < 4 

14. Example – Solve –0.5y < 6. Graph the solution set. 

–0.5y < 6 

(–2)( –0.5y) > (–2)(6) 

y > –12 

The solution set can be written {y|y > –12}. It is read the 

set of all numbers y such that y is greater than –12. 

This notation for solution sets is called 

set builder notation. 


15. Example – Solve 

x −11 

− x ≥ 

3 

–3x ≥ x – 11 

–4x ≥ – 11 

11 

x ≤ 4 

− x ≥ 

x −11 

. Graph the solution set. 

3 

11 

The solution set can be written {x|x ≤ }. 4 

16. The symbols ≠ ,≤, 

and ≥ can also be used when comparing numbers. The symbol 

≠ means is not equal to. The symbol ≤ means is less than or equal to. The symbol 

≥ means is greater than or equal to. 

6x ≠ 18 means 6x > 18 

or 6x < 18 

17. Example – Solve 3x > –27. Graph the solution set. 

3x > –27 

x > –9 

The solution set is {x|x > –9} 

18. Example – Solve –3x > 27. Graph the solution set. 

–3x > 27 

x < –9 

The solution set is {x|x < –9} 

19. Example – Solve 

3y 

− + 6 ≥ 3 

4 

3y 

− ≥ –3 

4 

y ≤ 4 

The solution set is {y|y ≤ 4} 

3y 

− + 6 ≥ 3. Graph the solution set. 

4 


20. Example – Solve 4a + 16 < –2(a + 4). Graph the solution set. 

4a + 16 < –2(a + 4) 

4a + 16 < –2a – 8 

6a < –24 

a < –4 

The solution set is {a|a < –4} 

21. The absolute value of a number represents its distance from zero on the number 

line. You can use this idea to help solve absolute value inequalities. 

22. Example – Solve |x| < 3. Graph the solution set. 

|x| < 3 means the distance between x 

and 0 is less than 3 units. To make |3| 

true, you must substitute values for x 

that are less than 3 units from 0. 

All the numbers between –3 and 3 

are less than three units from zero. 

The solution set is {x|–3< x < 3}. 

23. Example – Solve |x| ≥ 2. Graph the solution set. 

To make this true, you must 

substitute values for x that are 2 

or more units from 0. 

The solution set is {x|x ≥ 2 or 

x ≤ –2}. 

24. Example – Solve |2x – 5| > 9. Graph the solution set. 

The inequality |2x – 5| > 9 says that 2x – 5 is more than 9 units from 0. 

2x – 5 > 9 

2x > 14 

x > 7 

OR 

2x – 5 < –9 

2x < –4 

x < –2 The solution set is {x|x < –2 or x > 7} 


25. Example – Solve |8x| ≤ 24. Graph the solution set. 

The inequality |8x| ≤ 24 says that 8x is less than 24 units from 0. 

8x ≤ 24 

x ≤ 3 

AND 

8x ≥ –24 

x ≥ –3 

The solution set is {x|–3 ≤ x ≤ 3} 

26. Example – Solve |x + 2| > 5. Graph the solution set. 

The inequality |x + 2| > 5 says that x + 2 is more than 5 units from 0. 

x + 2 > 5 

x > 3 

OR 

x + 2 < –5 

x < –7 

The solution set is {x|x >3 or x < –7} 

27. Example – Solve |2x + 3| + 4 < 5. Graph the solution set. 

First rewrite the inequality by subtracting 4 from each side |2x + 3| < 1 

The inequality |2x + 3| < 1 says that 2x + 3 is less than 1 units from 0. 

2x + 3 < 1 

2x < –2 

x < –1 

AND 

2x + 3 > –1 

2x > –4 

x > –2 The solution set is {x|–2 < x < –1} 


28. Example – Solve |3x – 8| < 19. Graph the solution set. 

The inequality |3x – 8| < 19 says that 3x – 8 is less than 19 units from 0. 

3x – 8 < 19 

3x < 27 

x < 9 

AND 

3x – 8 > –19 

3x > –11 

11 

11 

x > − The solution set is {x| − < x < 9} 

3 

3 

29. Some absolute value inequalities have no solutions. For example, |4x – 9| < –7 is 

never true. Since the absolute value of a number is always positive or zero, there 

is not replacement for x that will make the sentence true. The inequality 

|4x – 9| < –7 has no solution. Therefore, its solution set is Ø. 

30. Some absolute value inequalities are always true. For example, |10x + 3| > –5 is 

always true. Since the absolute value of a number is always positive or zero, any 

replacement for x will make the sentence true. The solution set for |10x + 3| > –5 

is the set of real numbers. 

31. Example – Solve |6x + 2| + 5 > 3. Graph the solution set. 

First rewrite the inequality by subtracting 5 from each side 

|6x + 2| > –2 

The solution is all numbers since 

this statement is always true. 


32. Example – Solve |6x – 8| + 9 < 3. Graph the solution set. 

Rewrite the inequality by subtracting nine from each side 

|6x – 8| < –6 

There are no solutions since this 

statement is always false. 


Name:___________________ 

Date:____________ 

Class:___________________ 

Graph the solution set of each inequality. 

Solving Inequalities Worksheet 

1. x > –3 

2. a ≤ 0 

3. p ≥ 4 2 

1 

4. x < –7.5 

5. 

5 k ≤ 10 

6 

6. –3n > 6 

Solve each inequality. Graph the solution set. 

7. 3x + 7 > 43 

8. 7x – 5 ≥ 44 

9. 8 – 3x < 44 

10. x – 5 < 0.1 

12. 3x + 1 < x + 5 

13. 0.01x – 2.32 ≥ 0 

14. 

2 x + 3 

5 

< 0.03 

11. 5(3x + 2) > 7x – 2 

State an absolute value inequality for each of the following. Then graph each 

inequality. 

15. All numbers between –3 and 3. 

16. All numbers less than 8 and greater than –8. 

17. All numbers greater than 6 or less than –6. 

18. All numbers less than or equal to 5, and greater than or equal to –5. 

19. x > 3 or x < –3 

20. x < 6 and x > –6 

21. x ≤ 4 and x ≥ –4 


Solve each inequality. Graph each solution set. 

22. |x| < 9 

23. |x| ≥ 2 

24. |x + 1| > 3 

25. |3x| < 6 

26. |2x| ≥ –64 

27. |3x| < –15 

28. |6x + 25| + 14 < 6 

29. 6 + |3x| > 0 


Name:___________________ 

Date:____________ 

Class:___________________ 

Solving Inequalities Worksheet Key 

Graph the solution set of each inequality. 

1. x > –3 

2. a ≤ 0 

3. p ≥ 4 2 

1 

4. x < –7.5 

5. 

5 k ≤ 10 k ≤ 12 

6 

6. –3n > 6 n < –2 

Solve each inequality. Graph the solution set. 

7. 3x + 7 > 43 3x > 36 x > 12 

8. 7x – 5 ≥ 44 7x ≥ 49 x ≥ 7 

9. 8 – 3x < 44 – 3x < 36 x > –12 

10. x – 5 < 0.1 x < 5.1 


11. 5(3x + 2) > 7x – 2 15x + 10 > 7x – 2 8x > –12 x > 

1 

− 1 

2 

12. 3x + 1 < x + 5 2x < 4 x < 2 

13. 0.01x – 2.32 ≥ 0 0.01x ≥ 2.32 x ≥ 232 

14. 

2 x + 3 

5 

< 0.03 2x + 3 < 0.15 2x < –2.85 x < –1.425 

State an absolute value inequality for each of the following. Then graph each inequality. 

15. All numbers between –3 and 3. |x| < 3 

16. All numbers less than 8 and greater than –8. |x| > 8 

17. All numbers greater than 6 or less than –6. |x| > 6 

18. All numbers less than or equal to 5, and greater than or equal to –5. |x| ≤ 5 

19. x > 3 or x < –3 |x| > 3 


20. x < 6 and x > –6 |x| < 6 

21. x ≤ 4 and x ≥ –4 |x| ≤ 4 

Solve each inequality. Graph each solution set. 

22. |x| < 9 

x < 9 and x > –9 

23. |x| ≥ 2 

x ≤ –2 or x ≥ 2 

24. |x + 1| > 3 

x + 1 < –3 

x < –4 

or 

x + 1 > 3 

x > 2 

25. |3x| < 6 

3x > –6 

x > –2 

and 

3x < 6 

x < 2 

26. |2x| ≥ –64 

Since absolute values must be 

positive, all numbers work. 

27. |3x| < –15 

Since absolute values must be 

positive, no numbers work Ǿ 

28. |6x + 25| + 14 < 6 

First rewrite inequality by subtracting 6 from each side. 

(6x + 25| < –8 

Since absolute values must be positive, no 

numbers work Ǿ 


29. 6 + |3x| > 0 First rewrite inequality by subtracting 6 from each side. 

|3x| > –6 

Since absolute values must be positive, all numbers work. 


Student Name: __________________ 

Date: ______________ 

Solving Inequalities Checklist 

1. On questions 1 thru 6, did the student graph the inequality correctly? 

a. Yes (30 points) 

b. 5 out of 6 (25 points) 

c. 4 out of 6 (20 points) 

d. 3 out of 6 (15 points) 

e. 2 out of 6 (10 points) 

f. 1 out of 6 (5 points) 

2. On questions 7 thru 14, did the student solve the inequality correctly? 







g. 2 out of 8 (10 points) 

h. 1 out of 8 (5 points) 










4. On questions 15 thru 21, did the student state a correct absolute value for each inequality? 
















6. On questions 22 thru 29, did the student solve the inequality correctly? 









Total Number of Points _________ 

A 198 points and above 

B 176 points and above 

Any score below C 

needs 

remediation! 

C 154 points and above 

D 132 points and above 

F 

131 points and below 

NOTE: The sole purpose of this checklist is to aide the teacher 

in identifying students that need remediation. Students who 

meet the “C” criteria are ready for the next level of learning.

(a) Solving Inequalities Examples; (b) - Beacon Learning Center

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