Algebra I. Lesson 1, page 2

Let’s look at the following sentences.

1. England is in North America.

2. 2 + 7 = 10 - 1

3. The sum of 12 and 23 is greater than the sum of 13 and 34.

4. In an equilateral triangle, only the sides are equal.

5. 45 + 55 = (10)²

6. The area of a triangle whose two sides measure 12 and 16 is greater than 96

7. John is 5 feet 6 inches tall, Betty is 5 feet, 3 inches tall. What is the average height of John, Betty and Bill?

8. The condition x is greater than 0 is necessary in most problems in algebra.

Knowing the common definition of a sentence, we can say that all the above expressions are sentences. If we can determine a sentence is either true or false, it is then called a statement. We can explain the statements (1), (2) and (4)-(6) as follows:

1. We know that England is not in North America. Therefore, this statement is false.

2. The equality can be simplified as follows:
2 + 7 = 10 - 1
9 = 9

Therefore, the statement is true.

3. The sum of 12 and 23 is 35 and the sum of 13 and 34 is 47. 34 is less than 47. Therefore, the statement is false.

4. In any equilateral triangle, besides the sides, the interior angles are equal as well. Therefore, the statement is false.

5. The left side of the equality is equal to 100. The right side is 100 as well. Therefore, the statement is true.

Using different techniques, we could make decisions about the statements (1), (2), (4)-(6). Each statement is either true or false.

Statements (7), (8) may be either true or false. We cannot make any decision about their validity based on the given information. To do so, we need more information. Such expressions are called open sentences.

Example:

x + 2y = 45

A sum of some negative numbers is always greater than the sum of some integers
Maryanne is shorter than Adrianna.
His sister is older than he is.
The Empire State Building is the tallest building in the world.

Answer. All of the given expressions are open sentences except (5). We know that the Empire State Building is not the tallest building in the world. In the other expressions, we do not have enough data to make decisions on whether they are true or false.

Practice 2. Identify open sentences.

1) The area of a rectangle is greater than the area of a square.

2) x(x + 35) = 76

3) m + (2m + 23) = 23 + 3m

4) New York is the greatest city

5) x + 101 is less then 202

6) A triangle has more than two sides.

7) A rectangle has more than four sides.

8) All real numbers are either positive or negative.

Practice 3 (Application of Algebra in Geometry). Below are some shapes along with measures of some parts.

Using the given data, determine whether each expression in the table below is a false statement (F), true statement (T), or an open sentence (O).

Expression	Type
The area of the rectangle is 60
The perimeter of the rectangle is 34
The area of the parallelogram is 72.
The perimeter of the parallelogram is 16 + 2m. (m is the length of a lateral side)
The area of the right triangle is 88
The perimeter of the right triangle is 19 + x (x is the length of the hypotenuse)
The area of the square is 50
The perimeter of the square is greater than the perimeter of the rectangle.

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Open Sentences [25:35]

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Now let's do Practice Exercise 1-2 (top).

Next Page: Distributive and Additive Identity, Multiplicative Identity Property and Properties of Equality (top)