November 5, 2010

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 }


graph of a number line with all values less than 2 shaded in red
2.x/3 - 3 < -1
Answer: { x | x < 6 }


graph of a number line with all values less than 6 shaded in red
3.2x + 6 > 16
Answer: { x | x > 5 }


graph of a number line with all values greater than 5 shaded in red
4.x - 4 ≥ 9x - 60
Answer: { x | x ≤ 7 }


graph of a number line with all values less than or equal to 7 shaded in red
5.8x + 5 ≥ 3x + 10
Answer: { x | x ≥ 1 }


graph of a number line with all values greater than or equal to 1 shaded in red
6.2x - 3 > 9
Answer: { x | x > 6 }


graph of a number line with all values greater than 6 shaded in red
7.x - 8 > 6x - 13
Answer: { x | x < 1 }


graph of a number line with all values less than 1 shaded in red
8.4 - 5x > -6
Answer: { x | x < 2 }


graph of a number line with all values less than 2 shaded in red
9.4 - 16x > 8
Answer: { x | x < - 1/4 }


graph of a number line with all values less than - 1/4 shaded in red
10.x/2 + 7 > 9
Answer: { x | x > 4 }


graph of a number line with all values greater than 4 shaded in red
11.9x + 5 > 4x + 40
Answer: { x | x > 7 }


graph of a number line with all values greater than 7 shaded in red
12.8 - 5x < 5
Answer: { x | x > 3/5 }


graph of a number line with all values greater than 3/5 shaded in red
13.3x/4 + 2 > 8
Answer: { x | x > 8 }


graph of a number line with all values greater than 8 shaded in red
14.2x - 5 ≤ x + 1
Answer: { x | x ≤ 6 }


graph of a number line with all values less than or equal to 6 shaded in red
15.5x + 3 < 38
Answer: { x | x < 7 }


graph of a number line with all values less than 7 shaded in red
16.7 - 8x < 3
Answer: { x | x > 1/2 }


graph of a number line with all values greater than 1/2 shaded in red
17.4x + 3 ≤ 3x + 10
Answer: { x | x ≤ 7 }


graph of a number line with all values less than or equal to 7 shaded in red
18.5x + 7 < -8
Answer: { x | x < -3 }


graph of a number line with all values less than -3 shaded in red
19.2 - 5x < -33
Answer: { x | x > 7 }


graph of a number line with all values greater than 7 shaded in red
20.2x - 5 > -21
Answer: { x | x > -8 }


graph of a number line with all values greater than -8 shaded in red
Posted by Gerardo Emerzona I