Lesson #6 Solving Linear Inequalities

An inequality is a comparison of two numbers which are not equal. There are two types of inequalities used in algebra: greater than (>) and less than (<). The statement "x is greater than 3" is written as "x > 3". The statement "w is less than 12" is written as "w < 12".

Solving linear inequalities is much like solving linear equations. The basic idea is to isolate the variable by moving all other terms and coefficients to the other side. While working on an inequality problem, you may do any of the following operations:

1) Multiply both sides by a positive number. It is especially useful to clear fractions from an inequality by multiplying both sides by the least common denominator.

2) Multiply through parenthesis groups and combine like terms on each side.

3) Add or subtract the same term on both sides.

4) Divide by any positive coefficient (the multiplier on the variable term).

 

Example 1. Solve the inequality 5x - 3 < 14.

Start to isolate the variable by moving the constant term (add 3 to both sides):
 

5x - 3 < 14

Add 3 to both sides:

+3 +3
 

5x < 17
 

 

Example 2. Solve the inequality 2 + 3(2y - 4) ³ 12.

Multiply through the parentheses:

2 + 3(2y - 4) ³ 12
 

2 + 6y - 12 ³ 12

Combine like terms:

6y - 10 ³ 12

Add 10 to both sides:

6y ³ 22

Divide by 6:

 

Example 3. Solve .

First, multiply by the least common

denominator to clear out the fractions.

8x + 30 £ 5x

Move variable terms to the same side:

-5x -5x
 

3x + 30 £ 0

Subtract 30 from both sides:

3x £ -30

Divide by 3:

x £ -10

 

Example 4. Solve 3(5w - 1) - 2(3w + 7) > 4(w - 2) + 6.

This is just like example 3, except it has more terms and parenthesis groups.

3(5w - 1) - 2(3w + 7) > 4(w - 2) + 6

15w - 3 - 6w - 14 > 4w - 8 + 6

9w - 17 > 4w - 2

-4w -4w

5w - 17 > -2

+ 17 +17

5w > 15

w > 3

 

There is one important difference between equations and inequalities: if you ever need to divide by a negative coefficient in the final step, you must also reverse the direction of the inequality sign.

 

Example 5. Solve -3x £ 27.

Divide both sides by -3 and reverse the inequality sign:

Thus x ³ -9.

 

Example 6. Solve 3(y + 1) - 2(5y - 3) > 34.

3y + 3 - 10y + 6 > 34

-7y + 9 > 34

-7y > 25

Exercises

Solve each inequality.

1. 2x - 3 > 12

2. 5x + 2 < 9

3. 10y - 8 ³ 20
     

4. 3z - 25 < 17

5. 2x + 1 > 3

6. 7w + 6 £ 13
     

7. 5(2x - 1) > 10

8. 2(3y + 2) < 8

9. 3(w + 2) + 5 > 6
     

10. 2(5x - 3) + 7 £ 12

11. 4(3 + 2x) - 4 ³ -5

12. 9(x - 1) - 3 < 2
   

13. 3(x - 3) - 2(x + 4) < 9

14. 5(x - 2) + 3(x + 1) ³ 4
   

15. 2(2y - 5) + 5(y - 3) > 1

16. 2(q - 4) + 6(q - 10) < 14
   

17. 10(x + 2) - 4(x - 5) ³ 40

18. 3(5h + 2) + 2(h - 8) £ 9
     

19. -7x < 14

20. -3y ³ 21

21. -5z > 20
     

22. -4t ³ -12

23. -3x ³ -48

24. -2y < 9
     

25. -9x + 3 < 7

26. -4x + 10 > 3

27. -10c - 12 £ -32
   

28. 2(x + 1) - 5(x - 7) ³ 3 + 2(x - 3)

29. 3(x - 2) + 4(x + 1) < x - 2(x + 5)
   

30. 7(y + 3) + y < 2(2y - 3) + 10

31. 4(3x - 8) - 2(2x + 1) £ 2(x - 1)
   

32. 2(z - 1) - 8(z - 3) < 3(z + 1) - 4(z - 2)

33. 15 - 4(3x - 1) ³ 2(x + 7) - 3(2x)

 

 

Check the answers to Lesson 6 exercises

Go on to Lesson 7

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