# 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__**:**

- Correct Sentences (True Sentences)
- Incorrect Sentences (False Sentences)
- 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:

- $$x + 5 = 8$$
- $$10 – x = 12$$
- $$2x – 10 = 6$$
- $$3x > 4$$
- $$4x + 5 > 15$$
- $$4 < 6 + 5$$

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.