The rules for solving inequalities are the same as for equations, except when dividing by a negative number, the inequality sign flips. This is demonstrated through examples of solving various types of inequalities, including those with fractions, variables on both sides, and using the distributive property. The key point is checking if the number divided by is positive or negative to determine if the inequality sign remains the same or flips.

1. The rules for solving inequalities
are the same as solving equations,
WITH ONE EXCEPTION:
IF YOU DIVIDE BY A NEGATIVE,
THE INEQUALITY FLIPS
Ex:
-4x > 12
-4 F -4
L
I
P
x < -3

2. Inequalities similar to unit 1 equations
2x + 6 < -10

3. Inequalities similar to unit 1 equations
2x + 6 < -10
-6 -6
2x < -10
Subtract 6

4. Inequalities similar to unit 1 equations
2x + 6 < -10
-6 -6
2x < -10
2
2
Divide by 2

5. Inequalities similar to unit 1 equations
2x + 6 < -10
-6 -6
2x < -10
2
2
x < -5
NOTE: The inequality DOES NOT FLIP
because we divided by a positive 2

6. Inequalities including the distributive property
3(x – 4) < 2

7. Inequalities including the distributive property
3(x – 4) < 2
3x – 12 < 2
distribute

8. Inequalities including the distributive property
3(x – 4) < 2
3x – 12 < 2
+12 +12
3x < 14
Add 12

9. Inequalities including the distributive property
3(x – 4) < 2
3x – 12 < 2
+12 +12
3x < 14
3
3
Divide by 3

10. Inequalities including the distributive property
3(x – 4) < 2
3x – 12 < 2
+12 +12
3x < 14
3
3
x < 14
3
NOTE: The inequality DOES NOT FLIP
because we divided by a positive 3

11. Inequalities including fractions
1
2
x+5<6

12. Inequalities including fractions
1
2
x+5<6
-5 -5
1 x<1
2
Subtract 5

13. Inequalities including fractions
1
2
x+5<6
-5 -5
1 x<1 2
2 2
1x < 2
Multiply by 2

14. Inequalities including fractions
1
2
x+5<6
-5 -5
1 x<1 2
2
2
1x < 2
1 1
Divide by 1

15. Inequalities including fractions
1
2
x+5<6
-5 -5
1 x<1 2
2
2
1x < 2
1 1
x<2
NOTE: The inequality DOES NOT FLIP
because we divided by a positive 1

16. Inequalities with variables on both sides
3x – 8 < 5x + 2

17. Inequalities with variables on both sides
3x – 8 < 5x + 2
-5x
-5x
-2x – 8 < 2
Subtract 5x

18. Inequalities with variables on both sides
3x – 8 < 5x + 2
-5x
-5x
-2x – 8 < 2
+8 +8 Add 8
-2x < 10

19. Inequalities with variables on both sides
3x – 8 < 5x + 2
-5x
-5x
-2x – 8 < 2
+8 +8
-2x < 10 Divide by NEGATIVE 2
-2
-2

20. Inequalities with variables on both sides
3x – 8 < 5x + 2
-5x
-5x
-2x – 8 < 2
+8 +8
-2x < 10
-2
-2
x > -5

21. Inequalities with variables on both sides
3x – 8 < 5x + 2
-5x
-5x
-2x – 8 < 2
+8 +8
-2x < 10
-2
-2
x > -5
NOTE: The inequality FLIPS
because we divided by a NEGATIVE 2

22. Inequalities with distributive property AND variables on both sides
-2(2x – 3) > -8(x + 1)

23. Inequalities with distributive property AND variables on both sides
-2(2x – 3) > -8(x + 1)
-4x + 6 > -8x – 8
Distribute first!

24. Inequalities with distributive property AND variables on both sides
-2(2x – 3) > -8(x + 1)
-4x + 6 > -8x – 8
Add 8x
+8x
+8x
4x + 6 >-8

25. Inequalities with distributive property AND variables on both sides
-2(2x – 3) > -8(x + 1)
-4x + 6 > -8x – 8
+8x
+8x
4x + 6 >-8
-6 -6 Subtract 6
4x > -14

26. Inequalities with distributive property AND variables on both sides
-2(2x – 3) > -8(x + 1)
-4x + 6 > -8x – 8
+8x
+8x
4x + 6 >-8
-6 -6
4x > -14 Divide by 4
4
4
-7
x> 2