# 1.1 Real numbers: algebra essentials  (Page 9/35)

 Page 9 / 35

Simplify each algebraic expression.

1. $\frac{2}{3}y-2\left(\frac{4}{3}y+z\right)$
2. $\frac{5}{t}-2-\frac{3}{t}+1$
3. $4p\left(q-1\right)+q\left(1-p\right)$
4. $9r-\left(s+2r\right)+\left(6-s\right)$
1. $\text{\hspace{0.17em}}\frac{2}{t}-1;$
2. $\text{\hspace{0.17em}}3pq-4p+q;$
3. $\text{\hspace{0.17em}}7r-2s+6$

## Simplifying a formula

A rectangle with length $\text{\hspace{0.17em}}L\text{\hspace{0.17em}}$ and width $\text{\hspace{0.17em}}W\text{\hspace{0.17em}}$ has a perimeter $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ given by $\text{\hspace{0.17em}}P=L+W+L+W.\text{\hspace{0.17em}}$ Simplify this expression.

If the amount $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ is deposited into an account paying simple interest $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ for time $\text{\hspace{0.17em}}t,$ the total value of the deposit $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is given by $\text{\hspace{0.17em}}A=P+Prt.\text{\hspace{0.17em}}$ Simplify the expression. (This formula will be explored in more detail later in the course.)

$A=P\left(1+rt\right)$

Access these online resources for additional instruction and practice with real numbers.

## Key concepts

• Rational numbers may be written as fractions or terminating or repeating decimals. See [link] and [link] .
• Determine whether a number is rational or irrational by writing it as a decimal. See [link] .
• The rational numbers and irrational numbers make up the set of real numbers. See [link] . A number can be classified as natural, whole, integer, rational, or irrational. See [link] .
• The order of operations is used to evaluate expressions. See [link] .
• The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties. See [link] .
• Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. See [link] . They take on a numerical value when evaluated by replacing variables with constants. See [link] , [link] , and [link]
• Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression. See [link] and [link] .

## Verbal

Is $\text{\hspace{0.17em}}\sqrt{2}\text{\hspace{0.17em}}$ an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

irrational number. The square root of two does not terminate, and it does not repeat a pattern. It cannot be written as a quotient of two integers, so it is irrational.

What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?

What do the Associative Properties allow us to do when following the order of operations? Explain your answer.

The Associative Properties state that the sum or product of multiple numbers can be grouped differently without affecting the result. This is because the same operation is performed (either addition or subtraction), so the terms can be re-ordered.

## Numeric

For the following exercises, simplify the given expression.

$10+2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}\left(5-3\right)$

$6÷2-\left(81÷{3}^{2}\right)$

$-6$

$18+{\left(6-8\right)}^{3}$

$-2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{\left[16÷{\left(8-4\right)}^{2}\right]}^{2}$

$-2$

$4-6+2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}7$

$3\left(5-8\right)$

$-9$

$4+6-10÷2$

$12÷\left(36÷9\right)+6$

9

${\left(4+5\right)}^{2}÷3$

$3-12\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2+19$

-2

$2+8\text{\hspace{0.17em}}×\text{\hspace{0.17em}}7÷4$

$5+\left(6+4\right)-11$

4

$9-18÷{3}^{2}$

$14\text{\hspace{0.17em}}×\text{\hspace{0.17em}}3÷7-6$

0

$9-\left(3+11\right)\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2$

$6+2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2-1$

9

$64÷\left(8+4\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2\right)$

$9+4\left({2}^{2}\right)$

25

${\left(12÷3\text{\hspace{0.17em}}×\text{\hspace{0.17em}}3\right)}^{2}$

$25÷{5}^{2}-7$

$-6$

$\left(15-7\right)\text{\hspace{0.17em}}×\text{\hspace{0.17em}}\left(3-7\right)$

$2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}4-9\left(-1\right)$

17

${4}^{2}-25\text{\hspace{0.17em}}×\text{\hspace{0.17em}}\frac{1}{5}$

$12\left(3-1\right)÷6$

4

## Algebraic

For the following exercises, solve for the variable.

$8\left(x+3\right)=64$

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has
how can we solve this problem
Sin(A+B) = sinBcosA+cosBsinA
Prove it
Eseka
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
write down the polynomial function with root 1/3,2,-3 with solution
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
what is the answer to dividing negative index
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
give me the waec 2019 questions