Algebraic Sentences

Algebraic Sentences

We know that algebraic sentences give the relation between two algebraic expressions. For example, in the sentence 2 6 = 8, it is stated that 2 6 and 8 have the same sense.

The following are a few examples of algebraic sentences:
1) a + 4 = 6
2) x - 4 \ne 5
3) K - M > 6
4) x - 5 > 6
5) a = 8
6) 2a > 4
7) 4a = 8
8) {a^2} = 9
9) 7x = 3x + 2x
10) 8x + 2 = 3x + 5x + 2

Types of Algebraic Sentences:

  1. Correct Sentences (True Sentences)
  2. Incorrect Sentences (False Sentences)
  3. Open Sentences

The Following are Correct (True) Sentences:

1) 2x = x + x
2) 6 + 2 = 8
3) 5x - 2x = 3x
4) 3b > 2b
5) 4x \ne 3x

The Following are Incorrect (False) Sentences:

1) 2 + 3 > 8
2) 2x > 3x
3) 7x = 3x + 2x
4) 7x + 2 \ne 3x + 4x + 2
5) 5x > 8x + 2

The Following are Open Sentences:

1) a + 5 = 6
2) 4x < 2x + 5
3) x - 3 > 7
4) x - 4 = 5
5) 2a\not > 6

All the above mentioned sentences are such that each one is true for some numerical value but not for all numerical values of variables. For example take a + 5 = 6; the sentence is true for a = 1 but not true for any other than 1. Hence, the sentences which are true for some numerical values of variables but not for all are called open sentences.

Equations and Inequalities:

Consider the following open sentences:

  1. x + 5 = 8
  2. 10 - x = 12
  3. 2x - 10 = 6
  4. 3x > 4
  5. 4x + 5 > 15
  6. 4 < 6 + 5

In (1), (2) and (3), the symbol  = is used. In other cases the symbols  > or  < are used. The sentences in which symbol  = is used are called equations, and when the symbol  < or  > is used, the sentences are called inequalities.

e.g. (4), (5) (6) are inequalities.