# 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.