Graphs of Linear Inequalities

In the following graphs, the shaded part of the number line (in red color) indicates the solution sets of the inequalities.

These problems, 20 in all, illustrate the following axioms (or postulates):

• Axiom 1 A quantity added to both sides of an inequality does not change the direction of inequality.
• Axiom 2 A quantity subtracted from both sides of an inequality does not change the direction of inequality.
• Axiom 3 Multiplying both sides of an inequality by a positive number does not change the direction of inequality.
• Axiom 4 Dividing both sides of an inequality by a positive number does not change the direction of inequality.
• Axiom 5 Multiplying both sides of an inequality by a negative number changes the direction of inequality.
• Axiom 6 Dividing both sides of an inequality by a negative number changes the direction of inequality.

Problems 4, 7, 8, 9, 12, 16 and 19 illustrate axioms 5 and 6.

The rest of the problems illustrate the remaining axioms.

1.	5x + 4 < 3x + 8
	Answer: { x | x < 2 }

2.	x/3 - 3 < -1
	Answer: { x | x < 6 }

3.	2x + 6 > 16
	Answer: { x | x > 5 }

4.	x - 4 ≥ 9x - 60
	Answer: { x | x ≤ 7 }

5.	8x + 5 ≥ 3x + 10
	Answer: { x | x ≥ 1 }

6.	2x - 3 > 9
	Answer: { x | x > 6 }

7.	x - 8 > 6x - 13
	Answer: { x | x < 1 }

8.	4 - 5x > -6
	Answer: { x | x < 2 }

9.	4 - 16x > 8
	Answer: { x | x < - 1/4 }

10.	x/2 + 7 > 9
	Answer: { x | x > 4 }

11.	9x + 5 > 4x + 40
	Answer: { x | x > 7 }

12.	8 - 5x < 5
	Answer: { x | x > 3/5 }

13.	3x/4 + 2 > 8
	Answer: { x | x > 8 }

14.	2x - 5 ≤ x + 1
	Answer: { x | x ≤ 6 }

15.	5x + 3 < 38
	Answer: { x | x < 7 }

16.	7 - 8x < 3
	Answer: { x | x > 1/2 }

17.	4x + 3 ≤ 3x + 10
	Answer: { x | x ≤ 7 }

18.	5x + 7 < -8
	Answer: { x | x < -3 }

19.	2 - 5x < -33
	Answer: { x | x > 7 }

20.	2x - 5 > -21
	Answer: { x | x > -8 }