2.1 Symbols and notations (Page 2/2)

Basic mathematics review Page 2 / 2

Equality and inequality symbols

\begin{array}{l} a = b & a and b are equal \\ a > b & a is strictly greater than b \\ a < b & a is strictly less than b \end{array}

Some variations of these symbols include

$\begin{array}{l} a \neq b & a is not equal to b \\ a \geq b & a is greater than or equal to b \\ a \leq b & a is less than or equal to b \end{array}$

The last five of the above symbols are inequality symbols. We can negate (change to the opposite) any of the above statements by drawing a line through the relation symbol (as in $a \neq b$ ), as shown below:

$a$ is not greater than $b$ can be expressed as either

$\begin{matrix} a > b & or & a \leq b. \end{matrix}$

$a$ is not less $b$ than can be expressed as either

$\begin{matrix} a < b & or & a \geq b. \end{matrix}$

$a < b$ and $a \geq b$ both indicate that $a$ is less than $b$ .

Grouping symbols

Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations are to be grouped together and considered as one number. The grouping symbols commonly used in algebra are

$\begin{array}{l} Parentheses: & (\begin{matrix} \end{matrix}) \\ Brackets: & [\begin{matrix} \end{matrix}] \\ Braces: & {\begin{matrix} \end{matrix}} \\ Bar: & \underline{\begin{matrix} \end{matrix}} \end{array}$

In a computation in which more than one operation is involved, grouping symbols help tell us which operations to perform first. If possible, we perform operations inside grouping symbols first.

Sample set b

$(4 + 17) - 6 = 21 - 6 = 15$

$8 (3 + 6) = 8 (9) = 72$

$5 [8 + (10 - 4)] = 5 [8 + 6] = 5 [14] = 70$

$2 {3 [4 (17 - 11)]} = 2 {3 [4 (6)]} = 2 {3 [24]} = 2 {72} = 144$

$\frac{9 (5 + 1)}{24 + 3}$ .

The fraction bar separates the two groups of numbers $9 (5 + 1)$ and $24 + 3$ . Perform the operations in the numerator and denominator separately.

$\frac{9 (5 + 1)}{24 + 3} = \frac{9 (6)}{24 + 3} = \frac{54}{24 + 3} = \frac{54}{27} = 2$

Practice set b

Use the grouping symbols to help perform the following operations.

$3 (1 + 8)$

$4 [2 (11 - 5)]$

$6 {2 [2 (10 - 9)]}$

$\frac{1 + 19}{2 + 3}$

The following examples show how to use algebraic notation to write each expression.

9 minus $y$ becomes $9 - y$

46 times $x$ becomes $46x$

7 times $(x + y)$ becomes $7 (x + y)$

4 divided by 3, times $z$ becomes $(\frac{4}{3}) z$

$(a - b)$ times $(b - a)$ divided by (2 times $a$ ) becomes $\frac{(a - b) (b - a)}{2 a}$

Introduce a variable ( any letter will do but here we’ll let $x$ represent the number) and use appropriate algebraic symbols to write the statement: A number plus 4 is strictly greater than 6. The answer is $x + 4 > 6$ .

The order of operations

Suppose we wish to find the value of $16 + 4 \cdot 9$ . We could

add 16 and 4, then multiply this sum by 9.
$16 + 4 \cdot 9 = 20 \cdot 9 = 180$
multiply 4 and 9, then add 16 to this product.
$16 + 4 \cdot 9 = 16 + 36 = 52$

We now have two values for one number. To determine the correct value we must use the standard order of operations .

Order of operations

Perform all operations inside grouping symbols, beginning with the innermost set.
Perform all multiplications and divisions, as you come to them, moving left-to-right.
Perform all additions and subtractions, as you come to them, moving left-to-right.

As we proceed in our study of algebra, we will come upon another operation, exponentiation, that will need to be inserted before multiplication and division. (See Section [link] .)

Sample set c

USe the order of operations to find the value of each number.

$\begin{array}{l} 16 + 4 \cdot 9 & Multiply first . \\ = 16 + 36 & Now add . \\ = 52 \end{array}$

$\begin{array}{l} (27 - 8) + 7 (6 + 12) & Combine within parentheses . \\ = 19 + 7 (18) & Multiply . \\ = 19 + 126 & Now add . \\ = 145 \end{array}$

$\begin{array}{l} 8 + 2 [4 + 3 (6 - 1)] & Begin with the innermost set of grouping symbols, (\begin{matrix} \end{matrix}) . \\ = 8 + 2 [4 + 3 (5)] & Now work within the next set of grouping symbols, [\begin{matrix} \end{matrix}] . \\ = 8 + 2 [4 + 15] \\ = 8 + 2 [19] \\ = 8 + 38 \\ = 46 \end{array}$

$\begin{array}{l} \frac{6 + 4 [2 + 3 (19 - 17)]}{18 - 2 [2 (3) + 2]} & = \frac{6 + 4 [2 + 3 (2)]}{18 - 2 [6 + 2]} \\ = \frac{6 + 4 [2 + 6]}{18 - 2 [8]} \\ = \frac{6 + 4 [8]}{18 - 16} \\ = \frac{6 + 32}{2} \\ = \frac{38}{2} \\ = 19 \end{array}$

Practice set c

Use the order of operations to find each value.

$25 + 8 (3)$

$2 + 3 (18 - 5 \cdot 2)$

$4 + 3 [2 + 3 (1 + 8 \div 4)]$

$\frac{19 + 2 {5 + 2 [18 + 6 (4 + 1)]}}{5 \cdot 6 - 3 (5) - 2}$

Exercises

For the following problems, use the order of operations to find each value.

$2 + 3 (6)$

$18 - 7 (8 - 3)$

$8 \cdot 4 \div 16 + 5$

$(21 + 4) \div 5 \cdot 2$

$3 (8 + 2) \div 6 + 3$

$6 (4 + 1) \div (16 \div 8) - 15$

$6 (4 - 1) + 8 (3 + 7) - 20$

$(8) (5) + 2 (14) + (1) (10)$

$61 - 22 + 4 [3 (10) + 11]$

203

$\frac{(1 + 16 - 3)}{7} + 5 (12)$

$\frac{8 (6 + 20)}{8} + \frac{3 (6 + 16)}{22}$

$18 \div 2 + 55$

$21 \div 7 \div 3$

$85 \div 5 \cdot 5 - 85$

$(300 - 25) \div (6 - 3)$

$91 \frac{2}{3}$

$4 \cdot 3 + 8 \cdot 28 - (3 + 17) + 11 (6)$

$2 {(7 + 7) + 6 [4 (8 + 2)]}$

508

$0 + 10 (0) + 15 [4 (3) + 1]$

$6.1 (2.2 + 1.8)$

$24.4$

$\frac{5.9}{2} + 0.6$

$(4 + 7) (8 - 3)$

$(10 + 5) (10 + 5) - 4 (60 - 4)$

$(\frac{5}{12} - \frac{1}{4}) + (\frac{1}{6} + \frac{2}{3})$

$4 (\frac{3}{5} - \frac{8}{15}) + 9 (\frac{1}{3} + \frac{1}{4})$

$\frac{0}{5} + \frac{0}{1} + 0 [2 + 4 (0)]$

$0 \cdot 9 + 4 \cdot 0 \div 7 + 0 [2 (2 - 2)]$

For the following problems, state whether the given statements are the same or different.

$\begin{matrix} x \geq y & and & x > y \end{matrix}$

different

$\begin{matrix} x = y & and & y = x \end{matrix}$

same

Represent the product of 3 and $x$ five different ways.

Represent the sum of $a$ and $b$ two different ways.

$a + b, b + a$

For the following problems, rewrite each phrase using algebraic notation.

Ten minus three

$x$ plus sixteen

$x + 16$

51 divided by $a$

81 times $x$

$81 x$

3 times $(x + y)$

$(x + b)$ times $(x + 7)$

$(x + b) (x + 7)$

3 times $x$ times $y$

$x$ divided by (7 times $b$ )

$\frac{x}{7 b}$

$(a + b)$ divided by $(a + 4)$

For the following problems, introduce a variable (any letter will do) and use appropriate algebraic symbols towrite the given statement.

A number minus eight equals seventeen.

$x - 8 = 17$

Five times a number, minus one, equals zero.

A number divided by six is greater than or equal to forty-four.

$\frac{x}{6} \geq 44$

Sixteen minus twice a number equals five.

Determine whether the statements for the following problems are true or false.

$6 - 4 (4) (1) \leq 10$

true

$5 (4 + 2 \cdot 10) \geq 110$

$8 \cdot 6 - 48 \leq 0$

true

$\frac{20 + 4.3}{16} < 5$

$2 [6 (1 + 4) - 8] > 3 (11 + 6)$

false

$6 [4 + 8 + 3 (26 - 15)]$ $3 [7 (10 - 4)]$

The number of different ways 5 people can be arranged in a row is $5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ . How many ways is this?

120

A box contains 10 computer chips. Three chips are to be chosen at random. The number of ways this can be done is

$\frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{3 \cdot 2 \cdot 1 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

How many ways is this?

The probability of obtaining four of a kind in a five-card poker hand is

$\frac{13 \cdot 48}{(52 \cdot 51 \cdot 50 \cdot 49 \cdot 48) \div (5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}$

What is this probability?

$0.00024, or \frac{1}{4165}$

Three people are on an elevator in a five story building. If each person randomly selects a floor on which to get off, the probability that at least two people get off on the same floor is

$1 - \frac{5 \cdot 4 \cdot 3}{5 \cdot 5 \cdot 5}$

What is this probability?

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Source: OpenStax, Basic mathematics review. OpenStax CNX. Jun 06, 2012 Download for free at http://cnx.org/content/col11427/1.2

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